Soliton dynamics in computational anatomy

Analysis of microRNA expression profiles dynamic in different life stages of Haemaphysalis longicornis ticks by deep sequencing of small RNA libraries

The three-host tick Haemaphysalis longicornis is an obligate blood-sucking ectoparasite. In life-stage transitions, microRNAs (miRNAs) show a variety of expression changes. To investigate these changes, deep sequencing technology was applied to identify the conserved and potentially novel miRNAs expressed during the different life stages of H. longicornis. Total RNA from eggs, unfed larvae, unfed nymphs and unfed adults was extracted for deep sequence analysis. Deep sequencing on a Hiseq 4000 generated a total of 111,192,069 reads, grouped into four small RNA (sRNA) libraries, one for each of the four developmental stages of H. longicornis. Among these sequences, 78 conserved and 55 potentially novel miRNAs were identified, including stage-specific and differentially expressed miRNAs. Gene ontology (GO) analysis indicated significantly enriched GO terms related to cell proliferation and differentiation, including specific terms for the processes of development, growth, metabolism, regulation of biological functions, reproduction, and membrane enzyme regular activity. Kyoto Encyclopedia of Gene and Genomes (KEGG) analysis revealed a significant enrichment of the insulin, notch, Hippo, and Wnt signaling pathways for growth and development. Our data highlight the abundance of miRNA changes (conserved and potentially novel) in the different life stages of H. longicornis. In particular, stage-specific miRNAs, as observed, are essential regulators for the development of H. longicornis.

PDE-constrained LDDMM via geodesic shooting and inexact Gauss-Newton-Krylov optimization using the incremental adjoint Jacobi equations

The class of non-rigid registration methods proposed in the framework of PDE-constrained large deformation diffeomorphic metric mapping is a particularly interesting family of physically meaningful diffeomorphic registration methods. Inexact Gauss-Newton-Krylov optimization has shown an excellent numerical accuracy and an extraordinarily fast convergence rate in this framework. However, the Galerkin representation of the non-stationary velocity fields does not provide proper geodesic paths. In this work, we propose a method for PDE-constrained LDDMM parameterized in the space of initial velocity fields under the EPDiff equation. The derivation of the gradient and the Hessian-vector products are performed on the final velocity field and transported backward using the adjoint and the incremental adjoint Jacobi equations. This way, we avoid the complex dependence on the initial velocity field in the computations. We also avoid the computation of the adjoint equation and its incremental counterpart that has been recently identified as a subtle problem in PDE-constrained LDDMM. The proposed method provides geodesics in the framework of PDE-constrained LDDMM, and it shows performance competing with benchmark PDE-constrained LDDMM and EPDiff-LDDMM methods.

Band-Limited Stokes Large Deformation Diffeomorphic Metric Mapping

The class of registration methods proposed in the framework of Stokes large deformation diffeomorphic metric mapping (LDDMM) is a particularly interesting family of physically meaningful diffeomorphic registration methods. Stokes-LDDMM methods are formulated as constrained variational problems, where the different physical models are imposed using the associated partial differential equations as hard constraints. The most significant limitation of Stokes-LDDMM framework is its huge computational complexity. The objective of this paper is to promote the use of Stokes-LDDMM in computational anatomy applications with an efficient approximation of the original variational problem. Thus, we propose a novel method for efficient Stokes-LDDMM diffeomorphic registration. Our method poses the constrained variational problem in the space of band-limited vector fields and it is implemented in the GPU. The performance of band-limited Stokes-LDDMM has been compared and evaluated with original Stokes-LDDMM, EPDiff-LDDMM, and band-limited EPDiff-LDDMM. The evaluation has been conducted in 3-D with the nonrigid image registration evaluation project database. Since the update equation in Stokes-LDDMM involves the action of low-pass filters, the computational complexity has been greatly alleviated with a modest accuracy lose. We have obtained a competitive performance for some method configurations. Overall, our proposed method may make feasible the extensive use of novel physically meaningful Stokes-LDDMM methods in different computational anatomy applications. In addition, our results reinforce the usefulness of band-limited vector fields in diffeomorphic registration methods involving the action of low-pass filters in the optimization, even in algorithmically challenging environments such as Stokes-LDDMM.

Variational principles for stochastic soliton dynamics

We develop a variational method of deriving stochastic partial differential equations whose solutions follow the flow of a stochastic vector field. As an example in one spatial dimension, we numerically simulate singular solutions (peakons) of the stochastically perturbed Camassa-Holm (CH) equation derived using this method. These numerical simulations show that peakon soliton solutions of the stochastically perturbed CH equation persist and provide an interesting laboratory for investigating the sensitivity and accuracy of adding stochasticity to finite dimensional solutions of stochastic partial differential equations. In particular, some choices of stochastic perturbations of the peakon dynamics by Wiener noise (canonical Hamiltonian stochastic deformations, CH-SD) allow peakons to interpenetrate and exchange order on the real line in overtaking collisions, although this behaviour does not occur for other choices of stochastic perturbations which preserve the Euler-Poincaré structure of the CH equation (parametric stochastic deformations, P-SD), and it also does not occur for peakon solutions of the unperturbed deterministic CH equation. The discussion raises issues about the science of stochastic deformations of finite-dimensional approximations of evolutionary partial differential equation and the sensitivity of the resulting solutions to the choices made in stochastic modelling.

Singular solutions of a modified two-component Camassa-Holm equation

The Camassa-Holm (CH) equation is a well-known integrable equation describing the velocity dynamics of shallow water waves. This equation exhibits spontaneous emergence of singular solutions (peakons) from smooth initial conditions. The CH equation has been recently extended to a two-component integrable system (CH2), which includes both velocity and density variables in the dynamics. Although possessing peakon solutions in the velocity, the CH2 equation does not admit singular solutions in the density profile. We modify the CH2 system to allow a dependence on the average density as well as the pointwise density. The modified CH2 system (MCH2) does admit peakon solutions in the velocity and average density. We analytically identify the steepening mechanism that allows the singular solutions to emerge from smooth spatially confined initial data. Numerical results for the MCH2 system are given and compared with the pure CH2 case. These numerics show that the modification in the MCH2 system to introduce the average density has little short-time effect on the emergent dynamical properties. However, an analytical and numerical study of pairwise peakon interactions for the MCH2 system shows a different asymptotic feature. Namely, besides the expected soliton scattering behavior seen in overtaking and head-on peakon collisions, the MCH2 system also allows the phase shift of the peakon collision to diverge in certain parameter regimes.

Evolutions equations in computational anatomy

One of the main purposes in computational anatomy is the measurement and statistical study of anatomical variations in organs, notably in the brain or the heart. Over the last decade, our group has progressively developed several approaches for this problem, all related to the Riemannian geometry of groups of diffeomorphisms and the shape spaces on which these groups act. Several important shape evolution equations that are now used routinely in applications have emerged over time. Our goal in this paper is to provide an overview of these equations, placing them in their theoretical context, and giving examples of applications in which they can be used. We introduce the required theoretical background before discussing several classes of equations of increasingly complexity. These equations include energy minimizing evolutions deriving from Riemannian gradient descent, geodesics, parallel transport and Jacobi fields.

Geodesic flows on semidirect-product Lie groups: geometry of singular measure-valued solutions

The EPDiff equation (or the dispersionless Camassa–Holm equation in one dimension) is a well-known example of geodesic motion on the Diff group of smooth invertible maps (diffeomorphisms). Its recent two-component extension governs geodesic motion on the semidirect product DiffⓈ , where denotes the space of scalar functions. This paper generalizes the second construction to consider geodesic motion on DiffⓈ , where denotes the space of scalar functions that take values on a certain Lie algebra (e.g. = ⊗ (3)). Measure-valued delta-like solutions are shown to be momentum maps possessing a dual pair structure, thereby extending previous results for the EPDiff equation. The collective Hamiltonians are shown to fit into the Kaluza–Klein theory of particles in a Yang–Mills field and these formulations are shown to apply also at the continuum partial differential equation level. In the continuum description, the Kaluza–Klein approach produces the Kelvin circulation theorem.

Transport of Relational Structures in Groups of Diffeomorphisms

This paper focuses on the issue of translating the relative variation of one shape with respect to another in a template centered representation. The context is the theory of Diffeomorphic Pattern Matching which provides a representation of the space of shapes of objects, including images and point sets, as an infinite dimensional Riemannian manifold which is acted upon by groups of diffeomorphisms. We discuss two main options for achieving our goal; the first one is the parallel translation, based on the Riemannian metric; the second one, based on the group action, is the coadjoint transport. These methods are illustrated with 3D experiments.

Generalized tensor-based morphometry of HIV/AIDS using multivariate statistics on deformation tensors

This paper investigates the performance of a new multivariate method for tensor-based morphometry (TBM). Statistics on Riemannian manifolds are developed that exploit the full information in deformation tensor fields. In TBM, multiple brain images are warped to a common neuroanatomical template via 3-D nonlinear registration; the resulting deformation fields are analyzed statistically to identify group differences in anatomy. Rather than study the Jacobian determinant (volume expansion factor) of these deformations, as is common, we retain the full deformation tensors and apply a manifold version of Hotelling's $T(2) test to them, in a Log-Euclidean domain. In 2-D and 3-D magnetic resonance imaging (MRI) data from 26 HIV/AIDS patients and 14 matched healthy subjects, we compared multivariate tensor analysis versus univariate tests of simpler tensor-derived indices: the Jacobian determinant, the trace, geodesic anisotropy, and eigenvalues of the deformation tensor, and the angle of rotation of its eigenvectors. We detected consistent, but more extensive patterns of structural abnormalities, with multivariate tests on the full tensor manifold. Their improved power was established by analyzing cumulative p-value plots using false discovery rate (FDR) methods, appropriately controlling for false positives. This increased detection sensitivity may empower drug trials and large-scale studies of disease that use tensor-based morphometry.

Multisymplectic formulation of fluid dynamics using the inverse map

We construct multisymplectic formulations of fluid dynamics using the inverse of the Lagrangian path map. This inverse map, the ‘back-to-labels’ map, gives the initial Lagrangian label of the fluid particle that currently occupies each Eulerian position. Explicitly enforcing the condition that the fluid particles carry their labels with the flow in Hamilton's principle leads to our multisymplectic formulation. We use the multisymplectic one-form to obtain conservation laws for energy, momentum and an infinite set of conservation laws arising from the particle relabelling symmetry and leading to Kelvin's circulation theorem. We discuss how multisymplectic numerical integrators naturally arise in this approach.

A hamiltonian particle method for diffeomorphic image registration

Diffeomorphic image registration, where images are aligned using diffeomorphic warps, is a popular subject for research in medical image analysis. We introduce a novel algorithm for computing diffeomorphic warps that solves the Euler equations on the diffeomorphism group explicitly, based on a discretisation of the Hamiltonian, rather than using an optimiser. The result is an algorithm that is many times faster than those considered previously.

Statistical properties of Jacobian maps and the realization of unbiased large-deformation nonlinear image registration

Maps of local tissue compression or expansion are often computed by comparing magnetic resonance imaging (MRI) scans using nonlinear image registration. The resulting changes are commonly analyzed using tensor-based morphometry to make inferences about anatomical differences, often based on the Jacobian map, which estimates local tissue gain or loss. Here, we provide rigorous mathematical analyses of the Jacobian maps, and use themto motivate a new numerical method to construct unbiased nonlinear image registration. First, we argue that logarithmic transformation is crucial for analyzing Jacobian values representing morphometric differences. We then examine the statistical distributions of log-Jacobian maps by defining the Kullback-Leibler (KL) distance on material density functions arising in continuum-mechanical models. With this framework, unbiased image registration can be constructed by quantifying the symmetric KL-distance between the identity map and the resulting deformation. Implementation details, addressing the proposed unbiased registration as well as the minimization of symmetric image matching functionals, are then discussed and shown to be applicable to other registration methods, such as inverse consistent registration. In the results section, we test the proposed framework, as well as present an illustrative application mapping detailed 3-D brain changes in sequential magnetic resonance imaging scans of a patient diagnosed with semantic dementia. Using permutation tests, we show that the symmetrization of image registration statistically reduces skewness in the log-Jacobian map.

Direct mapping of hippocampal surfaces with intrinsic shape context

We propose in this paper a new method for the mapping of hippocampal (HC) surfaces to establish correspondences between points on HC surfaces and enable localized HC shape analysis. A novel geometric feature, the intrinsic shape context, is defined to capture the global characteristics of the HC shapes. Based on this intrinsic feature, an automatic algorithm is developed to detect a set of landmark curves that are stable across population. The direct map between a source and target HC surface is then solved as the minimizer of a harmonic energy function defined on the source surface with landmark constraints. For numerical solutions, we compute the map with the approach of solving partial differential equations on implicit surfaces. The direct mapping method has the following properties: (1) it has the advantage of being automatic; (2) it is invariant to the pose of HC shapes. In our experiments, we apply the direct mapping method to study temporal changes of HC asymmetry in Alzheimer's disease (AD) using HC surfaces from 12 AD patients and 14 normal controls. Our results show that the AD group has a different trend in temporal changes of HC asymmetry than the group of normal controls. We also demonstrate the flexibility of the direct mapping method by applying it to construct spherical maps of HC surfaces. Spherical harmonics (SPHARM) analysis is then applied and it confirms our results on temporal changes of HC asymmetry in AD.

Lagrangian frame diffeomorphic image registration: Morphometric comparison of human and chimpanzee cortex

We develop a novel Lagrangian reference frame diffeomorphic image and landmark registration method. The algorithm uses the fixed Langrangian reference frame to define the map between coordinate systems, but also generates and stores the inverse map from the Eulerian to the Lagrangian frame. Computing both maps allows facile computation of both Eulerian and Langrangian quantities. We apply this algorithm to estimating a putative evolutionary change of coordinates between a population of chimpanzee and human cortices. Inter-species functional homologues fix the map explicitly, where they are known, while image similarities guide the alignment elsewhere. This map allows detailed study of the volumetric change between chimp and human cortex. Instead of basing the inter-species study on a single species atlas, we diffeomorphically connect the mean shape and intensity templates for each group. The human statistics then map diffeomorphically into the space of the chimpanzee cortex providing a comparison between species. The population statistics show a significant doubling of the relative prefrontal lobe size in humans, as compared to chimpanzees.

Statistics on diffeomorphisms via tangent space representations

In this paper, we present a linear setting for statistical analysis of shape and an optimization approach based on a recent derivation of a conservation of momentum law for the geodesics of diffeomorphic flow. Once a template is fixed, the space of initial momentum becomes an appropriate space for studying shape via geodesic flow since the flow at any point along the geodesic is completely determined by the momentum at the origin through geodesic shooting equations. The space of initial momentum provides a linear representation of the nonlinear diffeomorphic shape space in which linear statistical analysis can be applied. Specializing to the landmark matching problem of Computational Anatomy, we derive an algorithm for solving the variational problem with respect to the initial momentum and demonstrate principal component analysis (PCA) in this setting with three-dimensional face and hippocampus databases.