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We consider damped stochastic systems in a controlled (time-varying) potential and study their transition between specified Gibbs-equilibria states in finite time. By the second law of thermodynamics, the minimum amount of work needed to transition from one equilibrium state to another is the difference between the Helmholtz free energy of the two states and can only be achieved by a reversible (infinitely slow) process. The minimal gap between the work needed in a finite-time transition and the work during a reversible one, turns out to equal the square of the optimal mass transport (Wasserstein-2) distance between the two end-point distributions times the inverse of the duration needed for the transition. This result, in fact, relates non-equilibrium optimal control strategies (protocols) to gradient flows of entropy functionals via the Jordan-Kinderlehrer-Otto scheme. The purpose of this paper is to introduce ideas and results from the emerging field of stochastic thermodynamics in the setting of classical regulator theory, and to draw connections and derive such fundamental relations from a control perspective in a multivariable setting.

We consider damped stochastic systems in a controlled (time-varying) potential and study their transition between specified Gibbs-equilibria states in finite time. By the second law of thermodynamics, the minimum amount of work needed to transition from one equilibrium state to another is the difference between the Helmholtz free energy of the two states and can only be achieved by a reversible (infinitely slow) process. The minimal gap between the work needed in a finite-time transition and the work during a reversible one, turns out to equal the square of the optimal mass transport (Wasserstein-2) distance between the two end-point distributions times the inverse of the duration needed for the transition. This result, in fact, relates non-equilibrium optimal control strategies (protocols) to gradient flows of entropy functionals via the Jordan-Kinderlehrer-Otto scheme. The purpose of this paper is to introduce ideas and results from the emerging field of stochastic thermodynamics in the setting of classical regulator theory, and to draw connections and derive such fundamental relations from a control perspective in a multivariable setting.

We seek network routing towards a desired final distribution that can mediate possible random link failures. In other words, we seek a routing plan that utilizes alternative routes so as to be relatively robust to link failures. To this end, we provide a mathematical formulation of a relaxed transport problem where the final distribution only needs to be close to the desired one. The problem is cast as a maximum entropy problem for probability distributions on paths with an added terminal cost. The entropic regularizing penalty aims at distributing the choice of paths amongst possible alternatives. We prove that the unique solution may be obtained by solving a generalized Schrödinger system of equations. An iterative algorithm to compute the solution is provided. Each iteration of the algorithm contracts the distance (in the Hilbert metric) to the optimal solution by more than 1/2, leading to extremely fast convergence.

We introduce the problem of transporting vector-valued distributions. In this, a salient feature is that mass may ow between vectorial entries as well as across space (discrete or continuous). The theory relies on a first step taken to define an appropriate notion of optimal transport on a graph. The corresponding distance between distributions is readily computable via convex optimization and provides a suitable generalization of Wasserstein-type metrics. Building on this, we define Wasserstein-type metrics on vector-valued distributions supported on continuous spaces as well as graphs. Motivation for developing vector-valued mass transport is provided by applications such as color image processing, multimodality imaging, polarimetric radar, as well as network problems where resources may be vectorial.

We introduce an optimal mass transport framework on the space of Gaussian mixture models. These models are widely used in statistical inference. Specifically, we treat Gaussian mixture models as a submanifold of probability densities equipped with the Wasserstein metric. The topology induced by optimal transport is highly desirable and natural because, in contrast to total variation and other metrics, the Wasserstein metric is weakly continuous (i.e., convergence is equivalent to convergence of moments). Thus, our approach provides natural ways to compare, interpolate and average Gaussian mixture models. Moreover, the approach has low computational complexity. Different aspects of the framework are discussed and examples are presented for illustration purposes.

The classical Monge–Kantorovich (MK) problem as originally posed is concerned with how best to move a pile of soil or rubble to an excavation or fill with the least amount of work relative to some cost function. When the cost is given by the square of the Euclidean distance, one can define a metric on densities called the Wasserstein distance. In this note, we formulate a natural matrix counterpart of the MK problem for positive-definite density matrices. We prove a number of results about this metric including showing that it can be formulated as a convex optimisation problem, strong duality, an analogue of the Poincaré–Wirtinger inequality and a Lax–Hopf–Oleinik–type result. [ABSTRACT FROM AUTHOR]

Purpose: Current state‐of‐the‐art models for estimating the pharmacokinetic parameters do not account for intervoxel movement of the contrast agent (CA). We introduce an optimal mass transport (OMT) formulation that naturally handles intervoxel CA movement and distinguishes between advective and diffusive flows. Method: Ten patients with head and neck squamous cell carcinoma (HNSCC) were enrolled in the study between June 2014 and October 2015 and underwent DCE MRI imaging prior to beginning treatment. The CA tissue concentration information was taken as the input in the data‐driven OMT model. The OMT approach was tested on HNSCC DCE data that provides quantitative information for forward flux (ΦF) and backward flux (ΦB). OMT‐derived ΦF was compared with the volume transfer constant for CA, Ktrans, derived from the Extended Tofts Model (ETM). Results: The OMT‐derived flows showed a consistent jump in the CA diffusive behavior across the images in accordance with the known CA dynamics. The mean forward flux was 0.0082 ± 0.0091 (min-1) whereas the mean advective component was 0.0052 ± 0.0086 (min-1) in the HNSCC patients. The diffusive percentages in forward and backward flux ranged from 8.67% to 18.76% and 12.76% to 30.36%, respectively. The OMT model accounts for intervoxel CA movement and results show that the forward flux (ΦF) is comparable with the ETM‐derived Ktrans. Conclusions: This is a novel data‐driven study based on optimal mass transport principles applied to patient DCE imaging to analyze CA flow in HNSCC. [ABSTRACT FROM AUTHOR]

We propose unbalanced versions of the quantum mechanical version of optimal mass transport that is based on the Lindblad equation describing open quantum systems. One of them is a natural interpolation framework between matrices and matrix-valued measures via a quantum mechanical formulation of Fisher-Rao information and the matricial Wasserstein distance, and the second is an interpolation between Wasserstein distance and Frobenius norm. We also give analogous results for the matrix-valued density measures, i.e., we add a spatial dependency on the density matrices. This might extend the applications of the framework to interpolating matrix-valued densities/images with unequal masses. [ABSTRACT FROM AUTHOR]

We introduce the problem of transporting vector-valued distributions. In this, a salient feature is that mass may flow between vectorial entries as well as across space (discrete or continuous). The theory relies on a first step taken to define an appropriate notion of optimal transport on a graph. The corresponding distance between distributions is readily computable via convex optimization and provides a suitable generalization of Wasserstein-type metrics. Building on this, we define Wasserstein-type metrics on vector-valued distributions supported on continuous spaces as well as graphs. Motivation for developing vector-valued mass transport is provided by applications such as color image processing, multimodality imaging, polarimetric radar, as well as network problems where resources may be vectorial. [ABSTRACT FROM AUTHOR]

The development of reliable predictive models for individual cancer cell lines to identify an optimal cancer drug is a crucial step to accelerate personalized medicine, but vast differences in cancer cell lines and drug characteristics make it quite challenging to develop predictive models that result in high predictive power and explain the similarity of cell lines or drugs. Our study proposes a novel network-based methodology that breaks the problem into smaller, more interpretable problems to improve the predictive power of anti-cancer drug responses in cell lines. For the drug-sensitivity study, we used the GDSC database for 915 cell lines and 200 drugs. The theory of optimal mass transport was first used to separately cluster cell lines and drugs, using gene-expression profiles and extensive cheminformatic drug features, represented in a form of data networks. To predict cell-line specific drug responses, random forest regression modeling was separately performed for each cell-line drug cluster pair. Post-modeling biological analysis was further performed to identify potential biological correlates associated with drug responses. The network-based clustering method resulted in 30 distinct cell-line drug cluster pairs. Predictive modeling on each cell-line-drug cluster outperformed alternative computational methods in predicting drug responses. We found that among the four drugs top-ranked with respect to prediction performance, three targeted the PI3K/mTOR signaling pathway. Predictive modeling on clustered subsets of cell lines and drugs improved the prediction accuracy of cell-line specific drug responses. Post-modeling analysis identified plausible biological processes associated with drug responses. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel method for texture mapping of closed surfaces. Our method is based on the technique of optimal mass transport (also known as the "earth-mover's metric"). This is a classical problem that concerns determining the optimal way, in the sense of minimal transportation cost, of moving a pile of soil from one site to another. In our context, the resulting mapping is area preserving and minimizes angle distortion in the optimal mass sense. Indeed, we first begin with an angle-preserving mapping (which may greatly distort area) and then correct it using the mass transport procedure derived via a certain gradient flow. In order to obtain fast convergence to the optimal mapping, we incorporate a multiresolution scheme into our flow. We also use ideas from discrete exterior calculus in our computations. [ABSTRACT FROM AUTHOR]

In this paper we present a new computationally efficient numerical scheme for the minimizing flow approach for the computation of the optimal L2 mass transport mapping. In contrast to the integration of a time dependent partial differential equation proposed in [S. Angenent, S. Haker, and A. Tannenbaum, SIAM J. Math. Anal., 35 (2003), pp. 61-97], we employ in the present work a direct variational method. The efficacy of the approach is demonstrated on both real and synthetic data. [ABSTRACT FROM AUTHOR]

In this paper, we present two novel algorithms which produce flattened visualizations of branched physiological surfaces, such as vessels. The first approach is a conformal mapping algorithm based on the minimization of two Dirichlet functionals. From a triangulated representation of vessel surfaces, we show how the algorithm can be implemented using a finite element technique. The second method is an algorithm which adjusts the conformal mapping to produce a flattened representation of the original surface while preserving areas. This approach employs the theory of optimal mass transport. Furthermore, a new way of extracting center lines for vessel fly-throughs is provided. [ABSTRACT FROM AUTHOR]

The emerging field of radiomics that transforms standard-of-care images to quantifiable scalar statistics endeavors to reveal the information hidden in these macroscopic images. The concept of texture is widely used and essential in many radiomic-based studies. Practice usually reduces spatial multidimensional texture matrices, e.g., gray-level co-occurrence matrices (GLCMs), to summary scalar features. These statistical features have been demonstrated to be strongly correlated and tend to contribute redundant information; and does not account for the spatial information hidden in the multivariate texture matrices. This study proposes a novel pipeline to deal with spatial texture features in radiomic studies. A new set of textural features that preserve the spatial information inherent in GLCMs is proposed and used for classification purposes. The set of the new features uses the Wasserstein metric from optimal mass transport theory (OMT) to quantify the spatial similarity between samples within a given label class. In particular, based on a selected subset of texture GLCMs from the training cohort, we propose new representative spatial texture features, which we incorporate into a supervised image classification pipeline. The pipeline relies on the support vector machine (SVM) algorithm along with Bayesian optimization and the Wasserstein metric. The selection of the best GLCM references is considered for each classification label and is performed during the training phase of the SVM classifier using a Bayesian optimizer. We assume that sample fitness is defined based on closeness (in the sense of the Wasserstein metric) and high correlation (Spearman's rank sense) with other samples in the same class. Moreover, the newly defined spatial texture features consist of the Wasserstein distance between the optimally selected references and the remaining samples. We assessed the performance of the proposed classification pipeline in diagnosing the coronavirus disease 2019 (COVID-19) from computed tomographic (CT) images. To evaluate the proposed spatial features' added value, we compared the performance of the proposed classification pipeline with other SVM-based classifiers that account for different texture features, namely: statistical features only, optimized spatial features using Euclidean metric, non-optimized spatial features with Wasserstein metric. The proposed technique, which accounts for the optimized spatial texture feature with Wasserstein metric, shows great potential in classifying new COVID CT images that the algorithm has not seen in the training step. The MATLAB code of the proposed classification pipeline is made available. It can be used to find the best reference samples in other data cohorts, which can then be employed to build different prediction models. • Large data cohorts inherently have representative "reference" samples. • Radiomics statistical texture features lose the spatial information inherent in highdimensional texture matrices. • Proposing spatial texture features considering optimal reference samples. • Robust Wasserstein metric from Optimal Mass transport (OMT) theory used in the proposed classification pipeline. [ABSTRACT FROM AUTHOR]

The glymphatic pathway is a system which facilitates continuous cerebrospinal fluid (CSF) and interstitial fluid (ISF) exchange and plays a key role in removing waste products from the rodent brain. Dysfunction of the glymphatic pathway may be implicated in the pathophysiology of Alzheimer's disease. Intriguingly, the glymphatic system is most active during deep wave sleep general anesthesia. By using paramagnetic tracers administered into CSF of rodents, we previously showed the utility of MRI in characterizing a macroscopic whole brain view of glymphatic transport but we have yet to define and visualize the specific flow patterns. Here we have applied an alternative mathematical analysis approach to a dynamic time series of MRI images acquired every 4 min over ∼ 3 h in anesthetized rats, following administration of a small molecular weight paramagnetic tracer into the CSF reservoir of the cisterna magna. We use Optimal Mass Transport (OMT) to model the glymphatic flow vector field, and then analyze the flow to find the network of CSF-ISF flow channels. We use 3D visualization computational tools to visualize the OMT defined network of CSF-ISF flow channels in relation to anatomical and vascular key landmarks from the live rodent brain. The resulting OMT model of the glymphatic transport network agrees largely with the current understanding of the glymphatic transport patterns defined by dynamic contrast-enhanced MRI revealing key CSF transport pathways along the ventral surface of the brain with a trajectory towards the pineal gland, cerebellum, hypothalamus and olfactory bulb. In addition, the OMT analysis also revealed some interesting previously unnoticed behaviors regarding CSF transport involving parenchymal streamlines moving from ventral reservoirs towards the surface of the brain, olfactory bulb and large central veins. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a new computationally efficient numerical scheme for the minimizing flow approach for optimal mass transport (OMT) with applications to non-rigid 3D image registration. The approach utilizes all of the gray-scale data in both images, and the optimal mapping from image to image is the inverse of the optimal mapping from to . Further, no landmarks need to be specified, and the minimizer of the distance functional involved is unique. Our implementation also employs multigrid, and parallel methodologies on a consumer graphics processing unit (GPU) for fast computation. Although computing the optimal map has been shown to be computationally expensive in the past, we show that our approach is orders of magnitude faster then previous work and is capable of finding transport maps with optimality measures (mean curl) previously unattainable by other works (which directly influences the accuracy of registration). We give results where the algorithm was used to compute non-rigid registrations of 3D synthetic data as well as intra-patient pre-operative and post-operative 3D brain MRI datasets. [Copyright &y& Elsevier]