Part II: Imaging
This is the second of three parts. Parts I and III are here and here.
With the advent of soft tissue imaging technology--computed tomography (CT) in 1972 and magnetic resonance imaging (MRI) in 1977--it was possible to examine living brains. In 1982, physicians saw multiple sclerosis (MS) lesions for the first time in a live patient. Since then, clinicians have increasingly relied on brain scans for diagnosis and treatment. With this in vivo technology, disease progression in patients could be tracked by physicians and researchers, either individually, or as part of a longitudinal study of cohorts. This has led to a better understanding of white matter degenerative disease and has improved treatment options.
The MR signal can be assessed in different ways and the 1990s saw the emergence of two important MRI modalities. The first, Seiji Ogawa's 1990 proposal to use contrasts in blood oxygen response to map changes in brain activity, led to the development of functional magnetic resonance imaging (fMRI). The ability to view the brain in real time was a big step forward; it enabled us to study brain function and is responsible for the widespread use of fMRI among clinical neurologists, behavioral scientists, neuroscientists and others.
Diffusion tensor magnetic resonance imaging (DTI) was the second important MRI modality introduced. Water constitutes a big part of living tissue--white matter is 72% water--and the physical flow of fluid is described by a diffusion process. In 1994, Peter Basser, James Matiello and Denis Le Bihan, in a landmark paper, proposed a tensor model for diffusion where the flow of water was described by the magnitude and direction of the principal eigenvector at each image voxel. White matter fibers are inherently anisotropic and the first applications of DTI were studies of neural connectivity where fibers were tracked from end to end. Since the resolution of DTI is at the cellular level, it was also possible to detect disease--through indices such as fractional anisotropy (FA)--before it appeared in conventional MRI scans. Normal appearing white matter (NAWM) in MS is one example where compromised integrity manifests through lower FA values.
A second-order tensor model is adequate for DTI reproductions of coherent fiber tracks but in cases where fibers meet or cross, only one of these directions is retained. DTI tractography of callosal fibers, where the lateral projections are attenuated, is illustrative of this limitation. To overcome this shortcoming, high angular resolution diffusion imaging (HARDI) images acquired in several spatially uniform directions has been used. An orientation distribution function (ODF) that can model multiple maxima representing the different fiber directions replaces the simple tensor model at each voxel. HARDI datasets offer better resolution for important DTI applications such as connectivity studies and preoperative investigations.
Saturday, September 25, 2010
Saturday, September 18, 2010
Quantitative white matter fiber analysis: a short history (Part I)
Part I: Histological investigation
This is the first post in a three part series. Parts II and III are here and here.
The modern scientific study of white matter has its roots in the 19th century when links were being established between mental dysfunction and neuroanatomy. Correlations made between postmortem abnormalities in the brains of mental patients and clinical evaluations while they were living led to important discoveries. The identification of the arcuate fasciculus as a language pathway that connected the two language centers, the Broca and Wernicke regions, is one famous example. In that case, Carl Wernicke, who was developing language network models, made the association between lesions in the arcuate fasciculus and the various aphasias he had observed.
The impetus from these investigations crossed over to other developments. Theodor Meynert, the reputed neuroanatomist, had classified prominent white matter tracts or fasciculi, as they were known, based on the kinds of connections they made. Burdach and Déjérine published postmortem atlases, and both prominently included white matter dissections. New techniques for histopathological analysis were introduced. Notable among these was Camilio Golgi's staining method and Santiago Ramón y Cajal's use of it in his histolgical studies of nerve fibers.
Quantitative white matter fiber analysis benefited from these cumulative efforts which made studies in fiber thinning, demyelination and microstuctural damage possible. Today, postmortem dissections still give the most precise quantitative assessments.
Note of appreciation: This write-up was compiled based partly on Marco Catani's--I have pointed him out before--voluminous publications. He writes exceedingly well on the subject of language networks and related themes.
This is the first post in a three part series. Parts II and III are here and here.
The modern scientific study of white matter has its roots in the 19th century when links were being established between mental dysfunction and neuroanatomy. Correlations made between postmortem abnormalities in the brains of mental patients and clinical evaluations while they were living led to important discoveries. The identification of the arcuate fasciculus as a language pathway that connected the two language centers, the Broca and Wernicke regions, is one famous example. In that case, Carl Wernicke, who was developing language network models, made the association between lesions in the arcuate fasciculus and the various aphasias he had observed.
The impetus from these investigations crossed over to other developments. Theodor Meynert, the reputed neuroanatomist, had classified prominent white matter tracts or fasciculi, as they were known, based on the kinds of connections they made. Burdach and Déjérine published postmortem atlases, and both prominently included white matter dissections. New techniques for histopathological analysis were introduced. Notable among these was Camilio Golgi's staining method and Santiago Ramón y Cajal's use of it in his histolgical studies of nerve fibers.
Quantitative white matter fiber analysis benefited from these cumulative efforts which made studies in fiber thinning, demyelination and microstuctural damage possible. Today, postmortem dissections still give the most precise quantitative assessments.
Note of appreciation: This write-up was compiled based partly on Marco Catani's--I have pointed him out before--voluminous publications. He writes exceedingly well on the subject of language networks and related themes.
Tuesday, July 27, 2010
Saturday, July 24, 2010
To find a mean in a nonlinear manifold
These are some notes on the Karcher mean. I will be updating this post hopefully in the coming weeks.
In a Euclidean space, for a set of k points, x_1, x_2 ... x_k, the sample mean is:
In a nonlinear manifold, a simple summation is no longer possible. We can, however, make an extrinsic computation by embedding the manifold in a vector space, computing the Euclidean mean and projecting the result back into the manifold. A disadvantage of this approach is that the mean computed depends on the choice of embedding.
A second possibility is an intrinsic computation, i.e., one where we use intrinsic manifold computations to compute the mean.
To compute an intrinsic mean within a manifold, M, we use the concept of the mean as the centroid of a density. This idea was put forward by Fréchet to calculate means in a Riemanniann manifold. The computation involved a minimization but the existence and uniqueness of the resulting mean could not be guaranteed (see Pennec's 1999 NSIP paper for details). Karcher's proposal that a local instead of a global mean be used (see Karcher's 1977 paper), led to a practical implementation. We shall henceforth refer to this local mean as the Karcher Mean.
(To be updated ...)
Karcher Mean references I found helpful
Ricardo Ferreira et al. have a paper entitled Newton Method for Riemannian centroid computation in naturally reductive homogeneous spaces which has implementation details such as the intrinsic manifold computations for well known manifolds such as the sphere, the special orthogonal group, SO(n), and the space of positive definite matrices.
Bibliography
1) M. Fréchet, "Les elements aléatoires de nature quelconque dans un espace
distancié," Annales de l'Institut Henri Poincaré, Vol. 10, (1948) pp. 215-310.
2) X. Pennec, “Probabilities and statistics on Riemannian manifolds: Basic tools for geometric measurements,” in Proc. NSIP'99, Vol. 1, (1999), pp. 194–198.
3) H. Karcher, Riemannian center of mass and mollifier smoothing. Commun. Pure and Appl. Math. 30 (1977), pp. 509–541.
In a Euclidean space, for a set of k points, x_1, x_2 ... x_k, the sample mean is:
In a nonlinear manifold, a simple summation is no longer possible. We can, however, make an extrinsic computation by embedding the manifold in a vector space, computing the Euclidean mean and projecting the result back into the manifold. A disadvantage of this approach is that the mean computed depends on the choice of embedding.
A second possibility is an intrinsic computation, i.e., one where we use intrinsic manifold computations to compute the mean.
To compute an intrinsic mean within a manifold, M, we use the concept of the mean as the centroid of a density. This idea was put forward by Fréchet to calculate means in a Riemanniann manifold. The computation involved a minimization but the existence and uniqueness of the resulting mean could not be guaranteed (see Pennec's 1999 NSIP paper for details). Karcher's proposal that a local instead of a global mean be used (see Karcher's 1977 paper), led to a practical implementation. We shall henceforth refer to this local mean as the Karcher Mean.
(To be updated ...)
Karcher Mean references I found helpful
Ricardo Ferreira et al. have a paper entitled Newton Method for Riemannian centroid computation in naturally reductive homogeneous spaces which has implementation details such as the intrinsic manifold computations for well known manifolds such as the sphere, the special orthogonal group, SO(n), and the space of positive definite matrices.
Bibliography
1) M. Fréchet, "Les elements aléatoires de nature quelconque dans un espace
distancié," Annales de l'Institut Henri Poincaré, Vol. 10, (1948) pp. 215-310.
2) X. Pennec, “Probabilities and statistics on Riemannian manifolds: Basic tools for geometric measurements,” in Proc. NSIP'99, Vol. 1, (1999), pp. 194–198.
3) H. Karcher, Riemannian center of mass and mollifier smoothing. Commun. Pure and Appl. Math. 30 (1977), pp. 509–541.
Thursday, July 1, 2010
The corpus callosum and interhemispheric communication
The corpus callosum (CC), with over 300 million fibers, is the largest white matter fiber bundle in the human brain. (It is easily identifiable in conventional MRI scans.) Topographically, it is centered along the midsagittal plane with radiations that extend to the prefrontal and frontal cortex in the anterior brain, the sensory-motor cortex in the middle and the parietal, temporal and occipital lobes in the posterior half of the brain.
This large and heterogeneous collection of fibers is responsible for interhemispheric communication. Michael Gazzaniga, a neuroscientist at Dartmouth College, has being studying the nature of this left brain- right brain communication for over 30 years. Here he explains some of his fascinating findings to Alan Alda, former hawkeye, now host of Scientific American Frontiers. And this is one of Gazzaniga's papers. (A similar account of the mysterious workings of the brain first got me interested in brain imaging. The book in question was V.S. Ramachandran's Phantoms in the Brain.)
Because of the important role it plays, the CC is the focus of many studies. Some are concerned with changes in the shape, size or structure of the CC. These changes may occur due to aging or degenerative disease. On one end of the spectrum, work is being done to provide tools to measure and monitor these changes. At the other end are the clinical studies. An example of a clinical study might be one that links the different stages of the disease or aging process with physical alterations.
This large and heterogeneous collection of fibers is responsible for interhemispheric communication. Michael Gazzaniga, a neuroscientist at Dartmouth College, has being studying the nature of this left brain- right brain communication for over 30 years. Here he explains some of his fascinating findings to Alan Alda, former hawkeye, now host of Scientific American Frontiers. And this is one of Gazzaniga's papers. (A similar account of the mysterious workings of the brain first got me interested in brain imaging. The book in question was V.S. Ramachandran's Phantoms in the Brain.)
Because of the important role it plays, the CC is the focus of many studies. Some are concerned with changes in the shape, size or structure of the CC. These changes may occur due to aging or degenerative disease. On one end of the spectrum, work is being done to provide tools to measure and monitor these changes. At the other end are the clinical studies. An example of a clinical study might be one that links the different stages of the disease or aging process with physical alterations.
Tuesday, June 22, 2010
White Matter Fiber Analysis
Shape, scale, orientation and position, the physical features associated with white matter fibers, can, either individually or in combination, be used to define feature spaces designed for specific end-applications. Such a treatment is useful since the quantitative analysis of white matter fibers has diverse applications, each with a different focus and objective.
In recent work, we describe a Riemannian framework in which various combinations of these features are considered. (This was presented at the ISBI 2010 conference. The slides are here, a version of the paper here.)
The framework also provides tools for computing statistical summaries of curves which enables us to perform a full statistical analysis. In the context of DTI fibers, a mean and variance that describes the essential characteristics of the fiber bundle can be used to represent a set of fibers. We can then proceed to tasks of statistical inference such as parameter estimation and hypothesis testing.
I am currently using the tools and metrics defined within this mathematical framework to show how morphological changes due to disease progression can be studied. Shape distances in tandem with distances defined within other manifolds like the shape+orientation manifold give us very encouraging results.
In recent work, we describe a Riemannian framework in which various combinations of these features are considered. (This was presented at the ISBI 2010 conference. The slides are here, a version of the paper here.)
The framework also provides tools for computing statistical summaries of curves which enables us to perform a full statistical analysis. In the context of DTI fibers, a mean and variance that describes the essential characteristics of the fiber bundle can be used to represent a set of fibers. We can then proceed to tasks of statistical inference such as parameter estimation and hypothesis testing.
I am currently using the tools and metrics defined within this mathematical framework to show how morphological changes due to disease progression can be studied. Shape distances in tandem with distances defined within other manifolds like the shape+orientation manifold give us very encouraging results.
Wednesday, June 2, 2010
Differential geometry in 10 slides
Partha Niyogi's very lucid talk entitled Geometric Methods and Manifold Learning includes a brief and very basic introduction to differential geometry(starts at t=40:49) which I found helpful.
This was part of the Machine Learning Workshop I attended at the University of Chicago last June (MLSS'09). There were several other talks and tutorials of note. I especially enjoyed Emmanuel Candes' talk on sparse signal recovery. The talks are available at the videolectures website.
This was part of the Machine Learning Workshop I attended at the University of Chicago last June (MLSS'09). There were several other talks and tutorials of note. I especially enjoyed Emmanuel Candes' talk on sparse signal recovery. The talks are available at the videolectures website.
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