COMMIT: Convex Optimization Modeling for Microstructure Informed Tractography.

Tractography is a class of algorithms aiming at in vivo mapping the major neuronal pathways in the white matter from diffusion magnetic resonance imaging (MRI) data. These techniques offer a powerful tool to noninvasively investigate at the macroscopic scale the architecture of the neuronal connections of the brain. However, unfortunately, the reconstructions recovered with existing tractography algorithms are not really quantitative even though diffusion MRI is a quantitative modality by nature. As a matter of fact, several techniques have been proposed in recent years to estimate, at the voxel level, intrinsic microstructural features of the tissue, such as axonal density and diameter, by using multicompartment models. In this paper, we present a novel framework to reestablish the link between tractography and tissue microstructure. Starting from an input set of candidate fiber-tracts, which are estimated from the data using standard fiber-tracking techniques, we model the diffusion MRI signal in each voxel of the image as a linear combination of the restricted and hindered contributions generated in every location of the brain by these candidate tracts. Then, we seek for the global weight of each of them, i.e., the effective contribution or volume, such that they globally fit the measured signal at best. We demonstrate that these weights can be easily recovered by solving a global convex optimization problem and using efficient algorithms. The effectiveness of our approach has been evaluated both on a realistic phantom with known ground-truth and in vivo brain data. Results clearly demonstrate the benefits of the proposed formulation, opening new perspectives for a more quantitative and biologically plausible assessment of the structural connectivity of the brain.

Accelerated Microstructure Imaging via Convex Optimization (AMICO) from diffusion MRI data.

Microstructure imaging from diffusion magnetic resonance (MR) data represents an invaluable tool to study non-invasively the morphology of tissues and to provide a biological insight into their microstructural organization. In recent years, a variety of biophysical models have been proposed to associate particular patterns observed in the measured signal with specific microstructural properties of the neuronal tissue, such as axon diameter and fiber density. Despite very appealing results showing that the estimated microstructure indices agree very well with histological examinations, existing techniques require computationally very expensive non-linear procedures to fit the models to the data which, in practice, demand the use of powerful computer clusters for large-scale applications. In this work, we present a general framework for Accelerated Microstructure Imaging via Convex Optimization (AMICO) and show how to re-formulate this class of techniques as convenient linear systems which, then, can be efficiently solved using very fast algorithms. We demonstrate this linearization of the fitting problem for two specific models, i.e. ActiveAx and NODDI, providing a very attractive alternative for parameter estimation in those techniques; however, the AMICO framework is general and flexible enough to work also for the wider space of microstructure imaging methods. Results demonstrate that AMICO represents an effective means to accelerate the fit of existing techniques drastically (up to four orders of magnitude faster) while preserving accuracy and precision in the estimated model parameters (correlation above 0.9). We believe that the availability of such ultrafast algorithms will help to accelerate the spread of microstructure imaging to larger cohorts of patients and to study a wider spectrum of neurological disorders.

A convex optimization framework for global tractography

In this article we present a novel approach for diffusion MRI global tractography. Our formulation models the signal in each voxel as a linear combination of fiber-tract basis func- tions, which consist of a comprehensive set of plausible fiber tracts that are locally compatible with the measured MR signal. This large dictionary of candidate fibers is directly estimated from the data and, subsequently, efficient convex optimization techniques are used for recovering the smallest subset globally best fitting the measured signal. Experimen- tal results conducted on a realistic phantom demonstrate that our approach significantly reduces the computational cost of global tractography while still attaining a reconstruction quality at least as good as the state-of-the-art global methods.

Accelerated Microstructure Imaging via Convex Optimization (AMICO) in crossing fibers

Mapping the microstructure properties of the local tissues in the brain is crucial to understand any pathological condition from a biological perspective. Most of the existing techniques to estimate the microstructure of the white matter assume a single axon orientation whereas numerous regions of the brain actually present a fiber-crossing configuration. The purpose of the present study is to extend a recent convex optimization framework to recover microstructure parameters in regions with multiple fibers.

l0-deconvolution for compressive diffusion MRI

Diffusion MRI is a well established imaging modality providing a powerful way to non-invasively probe the structure of the white matter. Despite the potential of the technique, the intrinsic long scan times of these sequences have hampered their use in clinical practice. For this reason, a wide variety of methods have been proposed to shorten acquisition times. [...] We here review a recent work where we propose to further exploit the versatility of compressed sensing and convex optimization with the aim to characterize the fiber orientation distribution sparsity more optimally. We re-formulate the spherical deconvolution problem as a constrained l0 minimization.

Efficient total variation algorithm for fetal brain MRI reconstruction.

Fetal MRI reconstruction aims at finding a high-resolution image given a small set of low-resolution images. It is usually modeled as an inverse problem where the regularization term plays a central role in the reconstruction quality. Literature has considered several regularization terms s.a. Dirichlet/Laplacian energy, Total Variation (TV)- based energies and more recently non-local means. Although TV energies are quite attractive because of their ability in edge preservation, standard explicit steepest gradient techniques have been applied to optimize fetal-based TV energies. The main contribution of this work lies in the introduction of a well-posed TV algorithm from the point of view of convex optimization. Specifically, our proposed TV optimization algorithm or fetal reconstruction is optimal w.r.t. the asymptotic and iterative convergence speeds O(1/n2) and O(1/√ε), while existing techniques are in O(1/n2) and O(1/√ε). We apply our algorithm to (1) clinical newborn data, considered as ground truth, and (2) clinical fetal acquisitions. Our algorithm compares favorably with the literature in terms of speed and accuracy.

An efficient total variation algorithm for super-resolution in fetal brain MRI with adaptive regularization.

Although fetal anatomy can be adequately viewed in new multi-slice MR images, many critical limitations remain for quantitative data analysis. To this end, several research groups have recently developed advanced image processing methods, often denoted by super-resolution (SR) techniques, to reconstruct from a set of clinical low-resolution (LR) images, a high-resolution (HR) motion-free volume. It is usually modeled as an inverse problem where the regularization term plays a central role in the reconstruction quality. Literature has been quite attracted by Total Variation energies because of their ability in edge preserving but only standard explicit steepest gradient techniques have been applied for optimization. In a preliminary work, it has been shown that novel fast convex optimization techniques could be successfully applied to design an efficient Total Variation optimization algorithm for the super-resolution problem. In this work, two major contributions are presented. Firstly, we will briefly review the Bayesian and Variational dual formulations of current state-of-the-art methods dedicated to fetal MRI reconstruction. Secondly, we present an extensive quantitative evaluation of our SR algorithm previously introduced on both simulated fetal and real clinical data (with both normal and pathological subjects). Specifically, we study the robustness of regularization terms in front of residual registration errors and we also present a novel strategy for automatically select the weight of the regularization as regards the data fidelity term. Our results show that our TV implementation is highly robust in front of motion artifacts and that it offers the best trade-off between speed and accuracy for fetal MRI recovery as in comparison with state-of-the art methods.

Efficient total variation algorithm for fetal MRI reconstruction

Fetal MRI reconstruction aims at finding a high-resolution image given a small set of low-resolution images. It is usually modeled as an inverse problem where the regularization term plays a central role in the reconstruction quality. Literature has considered several regularization terms s.a. Dirichlet/Laplacian energy [1], Total Variation (TV)based energies [2,3] and more recently non-local means [4]. Although TV energies are quite attractive because of their ability in edge preservation, standard explicit steepest gradient techniques have been applied to optimize fetal-based TV energies. The main contribution of this work lies in the introduction of a well-posed TV algorithm from the point of view of convex optimization. Specifically, our proposed TV optimization algorithm for fetal reconstruction is optimal w.r.t. the asymptotic and iterative convergence speeds O(1/n(2)) and O(1/root epsilon), while existing techniques are in O(1/n) and O(1/epsilon). We apply our algorithm to (1) clinical newborn data, considered as ground truth, and (2) clinical fetal acquisitions. Our algorithm compares favorably with the literature in terms of speed and accuracy.

Three essays on liquidity risk

Preface In this thesis we study several questions related to transaction data measured at an individual level. The questions are addressed in three essays that will constitute this thesis. In the first essay we use tick-by-tick data to estimate non-parametrically the jump process of 37 big stocks traded on the Paris Stock Exchange, and of the CAC 40 index. We separate the total daily returns in three components (trading continuous, trading jump, and overnight), and we characterize each one of them. We estimate at the individual and index levels the contribution of each return component to the total daily variability. For the index, the contribution of jumps is smaller and it is compensated by the larger contribution of overnight returns. We test formally that individual stocks jump more frequently than the index, and that they do not respond independently to the arrive of news. Finally, we find that daily jumps are larger when their arrival rates are larger. At the contemporaneous level there is a strong negative correlation between the jump frequency and the trading activity measures. The second essay study the general properties of the trade- and volume-duration processes for two stocks traded on the Paris Stock Exchange. These two stocks correspond to a very illiquid stock and to a relatively liquid stock. We estimate a class of autoregressive gamma process with conditional distribution from the family of non-central gamma (up to a scale factor). This process was introduced by Gouriéroux and Jasiak and it is known as Autoregressive gamma process. We also evaluate the ability of the process to fit the data. For this purpose we use the Diebold, Gunther and Tay (1998) test; and the capacity of the model to reproduce the moments of the observed data, and the empirical serial correlation and the partial serial correlation functions. We establish that the model describes correctly the trade duration process of illiquid stocks, but have problems to adjust correctly the trade duration process of liquid stocks which present long-memory characteristics. When the model is adjusted to volume duration, it successfully fit the data. In the third essay we study the economic relevance of optimal liquidation strategies by calibrating a recent and realistic microstructure model with data from the Paris Stock Exchange. We distinguish the case of parameters which are constant through the day from time-varying ones. An optimization problem incorporating this realistic microstructure model is presented and solved. Our model endogenizes the number of trades required before the position is liquidated. A comparative static exercise demonstrates the realism of our model. We find that a sell decision taken in the morning will be liquidated by the early afternoon. If price impacts increase over the day, the liquidation will take place more rapidly.

Large-scale clustering through functional embedding

We present a new framework for large-scale data clustering. The main idea is to modify functional dimensionality reduction techniques to directly optimize over discrete labels using stochastic gradient descent. Compared to methods like spectral clustering our approach solves a single optimization problem, rather than an ad-hoc two-stage optimization approach, does not require a matrix inversion, can easily encode prior knowledge in the set of implementable functions, and does not have an ?out-of-sample? problem. Experimental results on both artificial and real-world datasets show the usefulness of our approach.

Surface reconstruction from microscopic images in optical lithography.

This paper presents a method to reconstruct 3D surfaces of silicon wafers from 2D images of printed circuits taken with a scanning electron microscope. Our reconstruction method combines the physical model of the optical acquisition system with prior knowledge about the shapes of the patterns in the circuit; the result is a shape-from-shading technique with a shape prior. The reconstruction of the surface is formulated as an optimization problem with an objective functional that combines a data-fidelity term on the microscopic image with two prior terms on the surface. The data term models the acquisition system through the irradiance equation characteristic of the microscope; the first prior is a smoothness penalty on the reconstructed surface, and the second prior constrains the shape of the surface to agree with the expected shape of the pattern in the circuit. In order to account for the variability of the manufacturing process, this second prior includes a deformation field that allows a nonlinear elastic deformation between the expected pattern and the reconstructed surface. As a result, the minimization problem has two unknowns, and the reconstruction method provides two outputs: 1) a reconstructed surface and 2) a deformation field. The reconstructed surface is derived from the shading observed in the image and the prior knowledge about the pattern in the circuit, while the deformation field produces a mapping between the expected shape and the reconstructed surface that provides a measure of deviation between the circuit design models and the real manufacturing process.

On evaluating the accuracy and biological plausibility of diffusion MRI tractograms

In diffusion MRI, traditional tractography algorithms do not recover truly quantitative tractograms and the structural connectivity has to be estimated indirectly by counting the number of fiber tracts or averaging scalar maps along them. Recently, global and efficient methods have emerged to estimate more quantitative tractograms by combining tractography with local models for the diffusion signal, like the Convex Optimization Modeling for Microstructure Informed Tractography (COMMIT) framework. In this abstract, we show the importance of using both (i) proper multi-compartment diffusion models and (ii) adequate multi-shell acquisitions, in order to evaluate the accuracy and the biological plausibility of the tractograms.

A musculoskeletal shoulder model based on pseudo-inverse and null-space optimization.

The goal of the present work was assess the feasibility of using a pseudo-inverse and null-space optimization approach in the modeling of the shoulder biomechanics. The method was applied to a simplified musculoskeletal shoulder model. The mechanical system consisted in the arm, and the external forces were the arm weight, 6 scapulo-humeral muscles and the reaction at the glenohumeral joint, which was considered as a spherical joint. The muscle wrapping was considered around the humeral head assumed spherical. The dynamical equations were solved in a Lagrangian approach. The mathematical redundancy of the mechanical system was solved in two steps: a pseudo-inverse optimization to minimize the square of the muscle stress and a null-space optimization to restrict the muscle force to physiological limits. Several movements were simulated. The mathematical and numerical aspects of the constrained redundancy problem were efficiently solved by the proposed method. The prediction of muscle moment arms was consistent with cadaveric measurements and the joint reaction force was consistent with in vivo measurements. This preliminary work demonstrated that the developed algorithm has a great potential for more complex musculoskeletal modeling of the shoulder joint. In particular it could be further applied to a non-spherical joint model, allowing for the natural translation of the humeral head in the glenoid fossa.

Combination of enteral and parenteral nutrition in intensive care patients: A new concept of optimization

Undernutrition is a widespread problem in intensive care unit and is associated with a worse clinical outcome. A state of negative energy balance increases stress catabolism and is associated with increased morbidity and mortality in ICU patients. Undernutrition-related increased morbidity is correlated with an increase in the length of hospital stay and health care costs. Enteral nutrition is the recommended feeding route in critically ill patients, but it is often insufficient to cover the nutritional needs. The initiation of supplemental parenteral nutrition, when enteral nutrition is insufficient, could optimize the nutritional therapy by preventing the onset of early energy deficiency, and thus, could allow to reduce morbidity, length of stay and costs, shorten recovery period and, finally, improve quality of life. (C) 2009 Elsevier Masson SAS. All rights reserved.

Fitness, inclusive fitness, and optimization

Individual-as-maximizing agent analogies result in a simple understanding of the functioning of the biological world. Identifying the conditions under which individuals can be regarded as fitness maximizing agents is thus of considerable interest to biologists. Here, we compare different concepts of fitness maximization, and discuss within a single framework the relationship between Hamilton's (J Theor Biol 7: 1-16, 1964) model of social interactions, Grafen's (J Evol Biol 20: 1243-1254, 2007a) formal Darwinism project, and the idea of evolutionary stable strategies. We distinguish cases where phenotypic effects are additive separable or not, the latter not being covered by Grafen's analysis. In both cases it is possible to define a maximand, in the form of an objective function phi(z), whose argument is the phenotype of an individual and whose derivative is proportional to Hamilton's inclusive fitness effect. However, this maximand can be identified with the expression for fecundity or fitness only in the case of additive separable phenotypic effects, making individual-as-maximizing agent analogies unattractive (although formally correct) under general situations of social interactions. We also feel that there is an inconsistency in Grafen's characterization of the solution of his maximization program by use of inclusive fitness arguments. His results are in conflict with those on evolutionary stable strategies obtained by applying inclusive fitness theory, and can be repaired only by changing the definition of the problem.