Charles L. Epstein

Symmetric Diffeomorphic Image Registration with Cross-Correlation: Evaluating Automated Labeling of Elderly and Neurodegenerative Brain

One of the most challenging problems in modern neuroimaging is detailed characterization of neurodegeneration. Quantifying spatial and longitudinal atrophy patterns is an important component of this process. These spatiotemporal signals will aid in discriminating between related diseases, such as frontotemporal dementia (FTD) and Alzheimers disease (AD), which manifest themselves in the same at-risk population. Here, we develop a novel symmetric image normalization method (SyN) for maximizing the cross-correlation within the space of diffeomorphic maps and provide the Euler-Lagrange equations necessary for this optimization. We then turn to a careful evaluation of our method. Our evaluation uses gold standard, human cortical segmentation to contrast SyNs performance with a related elastic method and with the standard ITK implementation of Thirions Demons algorithm. The new method compares favorably with both approaches, in particular when the distance between the template brain and the target brain is large. We then report the correlation of volumes gained by algorithmic cortical labelings of FTD and control subjects with those gained by the manual rater. This comparison shows that, of the three methods tested, SyNs volume measurements are the most strongly correlated with volume measurements gained by expert labeling. This study indicates that SyN, with cross-correlation, is a reliable method for normalizing and making anatomical measurements in volumetric MRI of patients and at-risk elderly individuals.

The Bad Truth about Laplace's Transform

Inverting the Laplace transform is a paradigm for exponentially ill-posed problems. For a class of operators, including the Laplace transform, we give forward and inverse formulae that have fast implementations using the fast Fourier transform. These formulae lead easily to regularized inverses whose effects on noise and filtered data can be precisely described. Our results give cogent reasons for the general sense of dread most mathematicians feel about inverting the Laplace transform.

Accuracy and precision of MR blood oximetry based on the long paramagnetic cylinder approximation of large vessels

An accurate noninvasive method to measure the hemoglobin oxygen saturation (%HbO2) of deep‐lying vessels without catheterization would have many clinical applications. Quantitative MRI may be the only imaging modality that can address this difficult and important problem. MR susceptometry–based oximetry for measuring blood oxygen saturation in large vessels models the vessel as a long paramagnetic cylinder immersed in an external field. The intravascular magnetic susceptibility relative to surrounding muscle tissue is a function of oxygenated hemoglobin (HbO2) and can be quantified with a field‐mapping pulse sequence. In this work, the methods accuracy and precision was investigated theoretically on the basis of an analytical expression for the arbitrarily oriented cylinder, as well as experimentally in phantoms and in vivo in the femoral artery and vein at 3T field strength. Errors resulting from vessel tilt, noncircularity of vessel cross‐section, and induced magnetic field gradients were evaluated and methods for correction were designed and implemented. Hemoglobin saturation was measured at successive vessel segments, differing in geometry, such as eccentricity and vessel tilt, but constant blood oxygen saturation levels, as a means to evaluate measurement consistency. The average standard error and coefficient of variation of measurements in phantoms were <2% with tilt correction alone, in agreement with theory, suggesting that high accuracy and reproducibility can be achieved while ignoring noncircularity for tilt angles up to about 30°. In vivo, repeated measurements of %HbO2 in the femoral vessels yielded a coefficient of variation of less than 5%. In conclusion, the data suggest that %HbO2 can be measured reproducibly in vivo in large vessels of the peripheral circulation on the basis of the paramagnetic cylinder approximation of the incremental field. Magn Reson Med, 2009.

Generalized population models and the nature of genetic drift

The Wright-Fisher model of allele dynamics forms the basis for most theoretical and applied research in population genetics. Our understanding of genetic drift, and its role in suppressing the deterministic forces of Darwinian selection has relied on the specific form of sampling inherent to the Wright-Fisher model and its diffusion limit. Here we introduce and analyze a broad class of forward-time population models that share the same mean and variance as the Wright-Fisher model, but may otherwise differ. The proposed class unifies and further generalizes a number of population-genetic processes of recent interest, including the Λ and Cannings processes. Even though these models all have the same variance effective population size, they encode a rich diversity of alternative forms of genetic drift, with significant consequences for allele dynamics. We characterize in detail the behavior of standard population-genetic quantities across this family of generalized models. Some quantities, such as heterozygosity, remain unchanged; but others, such as neutral absorption times and fixation probabilities under selection, deviate by orders of magnitude from the Wright-Fisher model. We show that generalized population models can produce startling phenomena that differ qualitatively from classical behavior - such as assured fixation of a new mutant despite the presence of genetic drift. We derive the forward-time continuum limits of the generalized processes, analogous to Kimuras diffusion limit of the Wright-Fisher process, and we discuss their relationships to the Kingman and non-Kingman coalescents. Finally, we demonstrate that some non-diffusive, generalized models are more likely, in certain respects, than the Wright-Fisher model itself, given empirical data from Drosophila populations.

WRIGHT―FISHER DIFFUSION IN ONE DIMENSION

We analyze the diffusion processes associated to equations of Wright-Fisher type in one spatial dimension. These are defined by a degenerate second order operator on the interval [0, 1], where the coefficient of the second order term vanishes simply at the endpoints, and the first order term is an inward-pointing vector field. We consider various aspects of this problem, motivated by applications in population genetics, including a sharp regularity theory for the zero flux boundary conditions, as well as a derivation of the precise asymptotics for solutions of this equation, both as t goes to 0 and infinity, and as x goes to 0, 1.

Accuracy of the cylinder approximation for susceptometric measurement of intravascular oxygen saturation

Susceptometry‐based MR oximetry has previously been shown suitable for quantifying hemoglobin oxygen saturation in large vessels for studying vascular reactivity and quantification of global cerebral metabolic rate of oxygen utilization. A key assumption underlying this method is that large vessels can be modeled as long paramagnetic cylinders. However, bifurcations, tapering, noncircular cross‐section, and curvature of these vessels produce substantial deviations from cylindrical geometry, which may lead to errors in hemoglobin oxygen saturation quantification. Here, the accuracy of the “long cylinder” approximation is evaluated via numerical computation of the induced magnetic field from 3D segmented renditions of three veins of interest (superior sagittal sinus, femoral and jugular vein). At a typical venous oxygen saturation of 65%, the absolute error in hemoglobin oxygen saturation estimated via a closed‐form cylinder approximation was 2.6% hemoglobin oxygen saturation averaged over three locations in the three veins studied and did not exceed 5% for vessel tilt angles <30° at any one location. In conclusion, the simulation results provide a significant level of confidence for the validity of the cylinder approximation underlying MR susceptometry‐based oximetry of large vessels. Magn Reson Med, 2012.

On the Convergence of Local Expansions of Layer Potentials

In a recently developed quadrature method (quadrature by expansion or QBX), it was demonstrated that weakly singular or singular layer potentials can be evaluated rapidly and accurately on-surface by making use of local expansions about carefully chosen off-surface points. In this paper, we derive estimates for the rate of convergence of these local expansions, providing the analytic foundation for the QBX method. The estimates may also be of mathematical interest, particularly for microlocal or asymptotic analysis in potential theory.

A global invariant for three dimensional CR-manifolds.

In this note we define a global, R-valued invariant of a compact, strictly pseudoconvex 3-dimensional CR-manifold M whose holomorphic tangent bundle is trivial. The invariant arises as the evaluation of a deRham cohomology class on the fundamental class of the manifold. To construct the relevant form, we start with the CR structure bundle Y over M (see [Ch-Mo], whose notation we follow). The form is a secondary characteristic form of this structure. By fixing a contact form and coframe, i.e., a section of Y, we obtain a form on M. Surprisingly, this form is well-defined up to an exact term, and thus its cohomology class is well-defined in H 3 (M, R). Our motivation for studying this invariant was its analogy with the R/Z secondary characteristic number associated by Chern and Simons to the conforreal class of a Riemannian 3-manifold N, which provides an obstruction to the conformal immersion of N in R 4. Though several formal analogies to the conformal case are valid for our invariant, this one does not hold up: specifically, in w below, we calculate examples which show that the CR invariant can take on any positive real value for hypersurfaces embedded in C 2. It is also clear that the invariant is neither a homotopy nor concordance invariant, but depends in an elusive way on the CR structure. Our inspiration came from the seminal papers of Chern and Moser and Chern and Simons. The idea of looking at secondary characteristic forms of higher order geometric structures in general appears in [Ko-Oc], though with a different intention. In w 2 we will quickly review the definition of a CR structure, the construction of y and its reduction to a pseudo-hermitian structure ~ ld Webster [-We]. In w 3 we define the invariant and prove that it is, in fact, R-valued, and not R/Z-valued as in the Riemannian case. We also prove that if the invariant is stationary as a function of the CR structure, then M is locally CR equivalent to the standard three sphere in C 2, paralleling a result of Chern and Simons. As noted already, w 4 is devoted to the calculation of several examples.

Dynamics of Neutral and Selected Alleles When the Offspring Distribution Is Skewed

We analyze the dynamics of two alternative alleles in a simple model of a population that allows for large family sizes in the distribution of offspring number. This population model was first introduced by Eldon and Wakeley, who described the backward-time genealogical relationships among sampled individuals, assuming neutrality. We study the corresponding forward-time dynamics of allele frequencies, with or without selection. We derive a continuum approximation, analogous to Kimura’s diffusion approximation, and we describe three distinct regimes of behavior that correspond to distinct regimes in the coalescent processes of Eldon and Wakeley. We demonstrate that the effect of selection is strongly amplified in the Eldon–Wakeley model, compared to the Wright–Fisher model with the same variance effective population size. Remarkably, an advantageous allele can even be guaranteed to fix in the Eldon–Wakeley model, despite the presence of genetic drift. We compute the selection coefficient required for such behavior in populations of Pacific oysters, based on estimates of their family sizes. Our analysis underscores that populations with the same effective population size may nevertheless experience radically different forms of genetic drift, depending on the reproductive mechanism, with significant consequences for the resulting allele dynamics.

Stability of embeddings for pseudoconcave surfaces and their boundaries

We consider the problem of projectively embedding pseudoconcave surfaces and the stability, under deformation of the complex structure of the algebra of meromorphic functions defined by a positively embedded divisor. We introduce weakened notions of embeddability for a smooth complex surface with strictly pseudoconcave boundary and a positively embedded, smooth divisor: “weak embeddability” and “almost embeddability.” Almost embeddability of a strictly pseudoconcave surface is shown to be equivalent to embeddability of its boundary. This allows us to establish, in many interesting examples that the set of embeddable CR–structures on the boundary is closed in the C∞–topology.