Infinite number of solutions to the hemodynamic inverse problem

Investigators have had much success solving the "hemodynamic forward problem," i.e., predicting pressure and flow at the entrance of an arterial system given knowledge of specific arterial properties and arterial system topology. Recently, the focus has turned to solving the "hemodynamic inverse problem," i.e., inferring mechanical properties of an arterial system from measured input pressure and flow. Conventional methods to solve the inverse problem rely on fitting to data simple models with parameters that represent specific mechanical properties. Controversies have arisen, because different models ascribe pressure and flow to different properties. However, an inherent assumption common to all model-based methods is the existence of a unique set of mechanical properties that yield a particular pressure and flow. The present work illustrates that there are, in fact, an infinite number of solutions to the hemodynamic inverse problem. Thus a measured pressure-flow pair can result from an infinite number of different arterial systems. Except for a few critical properties, conventional approaches to solve the inverse problem for specific arterial properties are futile.     

Dynamical properties of model gene networks and implications for the inverse problem

We study the inverse problem, or the "reverse-engineering" problem, for two abstract models of gene expression dynamics, discrete-time Boolean networks and continuous-time switching networks. Formally, the inverse problem is similar for both types of networks. For each gene, its regulators and its Boolean dynamics function must be identified. However, differences in the dynamical properties of these two types of networks affect the amount of data that is necessary for solving the inverse problem. We derive estimates for the average amounts of time series data required to solve the inverse problem for randomly generated Boolean and continuous-time switching networks. We also derive a lower bound on the amount of data needed that holds for both types of networks. We find that the amount of data required is logarithmic in the number of genes for Boolean networks, matching the general lower bound and previous theory, but are superlinear in the number of genes for continuous-time switching networks. We also find that the amount of data needed scales as 2(K), where K is the number of regulators per gene, rather than 2(2K), as previous theory suggests.     

EEG source localization: comparative study of classical and neural network methods.

We address the problem of estimating biopotential sources within the brain, based on EEG signals observed on the scalp. This problem, known as the inverse problem of electrophysiology, has no closed-form solution, and requires iterative techniques such as the Levenberg-Marquardt (LM) algorithm. Considering the nonlinear nature of the inverse problem, and the low signal to noise ratio inherent in EEG signals, a backpropagation neural network (BPN) has been recently proposed as a solution. The technique has not been properly compared with classical techniques such as the LM method, or with more recent neural network techniques such as the Radial Basis Function (RBF) network. In this paper, we provide improved strategies based on BPN and consider RBF networks in solving the inverse problem. We compare the performances of BPN, RBF and a hybrid technique with that of the classical LM method.     

A New Approach for Inversion of Large Random Matrices in Massive MIMO Systems

We report a novel approach for inversion of large random matrices in massive Multiple-Input Multiple Output (MIMO) systems. It is based on the concept of inverse vectors in which an inverse vector is defined for each column of the principal matrix. Such an inverse vector has to satisfy two constraints. Firstly, it has to be in the null-space of all the remaining columns. We call it the null-space problem. Secondly, it has to form a projection of value equal to one in the direction of selected column. We term it as the normalization problem. The process essentially decomposes the inversion problem and distributes it over columns. Each column can be thought of as a node in the network or a particle in a swarm seeking its own solution, the inverse vector, which lightens the computational load on it. Another benefit of this approach is its applicability to all three cases pertaining to a linear system: the fully-determined, the over-determined, and the under-determined case. It eliminates the need of forming the generalized inverse for the last two cases by providing a new way to solve the least squares problem and the Moore and Penrose''s pseudoinverse problem. The approach makes no assumption regarding the size, structure or sparsity of the matrix. This makes it fully applicable to much in vogue large random matrices arising in massive MIMO systems. Also, the null-space problem opens the door for a plethora of methods available in literature for null-space computation to enter the realm of matrix inversion. There is even a flexibility of finding an exact or approximate inverse depending on the null-space method employed. We employ the Householder''s null-space method for exact solution and present a complete exposition of the new approach. A detailed comparison with well-established matrix inversion methods in literature is also given.     

Inverse Problem in Materials Science

Abstract

The importance of the inverse problem in materials science, that is, the determination of unknown parameters in the physico-chemical models from the experimental values is emphasized. The main attention is given to the problem of how to take into account experimental systematic errors during the inverse problem.     

Inverse method of stress analysis for cerebral aneurysms

We present a method for predicting the wall stress in a class of cerebral aneurysms. The method hinges on an inverse formulation of the elastostatic equilibrium problem; it takes as the input a deformed configuration and the corresponding pressure, and predicts the wall stress in the given deformed state. For a membrane structure, the inverse formulation possesses a remarkable feature, that is, it can practically determine the wall tension without accurate knowledge of the wall elastic properties. In this paper, we present a finite element formulation for the inverse membrane problem and perform material sensitivity studies on idealized lesions and an image-based cerebral aneurysm model.     

Ameboid cell motility: a model and inverse problem, with an application to live cell imaging data

In this article a mathematical model for ameboid cell movement is developed using a spring-dashpot system with Newtonian dynamics. The model is based on the facts that the cytoskeleton plays a primary role for cell motility and that the cytoplasm is viscoelastic. Based on the model, the inverse problem can be posed: if a structure like a spring-dashpot system is embedded into the living cell, what kind of characteristic properties must the structure have in order to reproduce a given movement of the cell? This inverse problem is the primary topic of this paper. On one side the model mimics some features of the movement, and on the other side, the solution to the inverse problem provides model parameters that give some insight, principally into the mechanical aspect, but also, through qualitative reasoning, into chemical and biophysical aspects of the cell. Moreover, this analysis can be done locally or globally and in different media by using the simplest possible information: positions of the cell and nuclear membranes. It is shown that the model and solution to the inverse problem for simulated data sets are highly accurate. An application to a set of live cell imaging data obtained from random movements of a human brain tumor cell (U87-MG human glioblastoma cell line) then provides an example of the efficiency of the model, through the solution of its inverse problem, as a way of understanding experimental data.     

The inverse dynamics problem of neuromuscular control

The myoskeletal inverse dynamics problem and the myocybernetic control inverse problem were investigated with respect to their ill-posedness. The first problem consists of finding from observed experimental motion and reaction force data the resultant muscle moments that generated the observed motion, while the second aims at finding the corresponding neural controls. It is shown that both problems belong to the class of incorrectly posed (ill-posed) problems that, by definition, do not possess unique solutions. To illustrate this point, results of a forward dynamics simulation of a comprehensive neuromusculoskeletal model of the human body are presented. These results demonstrate that fairly chaotic neural control perturbations have very little influence on the resulting motion trajectory, at least in the present example. While a regularization procedure may be applied to solve successfully the myoskeletal inverse dynamics problem, the myocybernetic control inverse problem is unsolvable. The latter fact has the important implication that, based on the somatosensory inputs it receives, the pars intermedia in the cerebellum is not able to control individual motor unit stimulation rates and recruitment patterns but only whole muscles by means of a single compound signal. The latter signal is identified as the “common drive.” Presumably at the spinal level, special neural circuits are used to decompose the common drive signal into motor unit recruitment patterns and stimulation rates that are specific for a given mode of contraction and probably obey certain optimality principles. Received: 26 February 1999 / Accepted in revised form: 11 June 1999     

A model study of instability of the inverse problem in electrocardiography.

The inverse problem in electrocardiography is studied analytically using a concentric spheres model with no symmetry assumptions on the potential distribution. The mathematical formulation is presented, and existence and uniqueness of the solution are briefly discussed. Solution to the inverse problem is inherently very unstable. The magnitude of this instability is demonstrated using the derived analytical inverse solution for the spherical model. Regularization methods used to date are based on a regularization parameter that does not relate to any measurable physiological parameters. This paper presents a regularization method that is based on a parameter in the form of an a priori bound on the L2 norm of the inverse solution. Such a bound can be obtained from the theoretical estimates based on the measured values of the body surface potentials together with experimental knowledge about the magnitudes of the epicardial potentials. Based on the presented regularization, an exact form of the regularized solution and estimates of its accuracy are derived.     

A new method for regularization of the inverse problem of electrocardiography.

The inverse problem of electrocardiography (specifically, that part concerned with the computation of the ventricular surface activation isochrones) is shown to be formally equivalent to the problem of identification and measurement of discontinuities in derivatives of body surface potentials. This is based on the demonstration that such measurements allow localization of the relative extrema of the ventricular surface activation map (given a forward problem solution), which in turn restricts the space of admissible solution maps to a compact set. Although the inverse problem and the problem of identifying derivative discontinuities are both ill-posed, it is possible that the latter may be more easily or justifiably resolved with available information, particularly as current methods for regularizing the inverse problem typically rely on a regularization parameter chosen in an a posteriori fashion. An example of the power of the approach is the demonstration that a recent Uniform Dipole Layer Hypothesis-based method for producing the ventricular surface activation map is largely independent on that hypothesis and capable in principle of generating maps that are very similar in a precise sense to those that would result from the usual epicardial potential formulation (assuming the latter were capable of producing intrinsic deflections in computed epicardial electrograms sufficiently steep to accurately compute the activation map). This is consistent with the preliminary success of the former method, despite the significant inaccuracy of its underlying assumption.     

On the inverse problem of binocular 3D motion perception

It is shown that existing processing schemes of 3D motion perception such as interocular velocity difference, changing disparity over time, as well as joint encoding of motion and disparity, do not offer a general solution to the inverse optics problem of local binocular 3D motion. Instead we suggest that local velocity constraints in combination with binocular disparity and other depth cues provide a more flexible framework for the solution of the inverse problem. In the context of the aperture problem we derive predictions from two plausible default strategies: (1) the vector normal prefers slow motion in 3D whereas (2) the cyclopean average is based on slow motion in 2D. Predicting perceived motion directions for ambiguous line motion provides an opportunity to distinguish between these strategies of 3D motion processing. Our theoretical results suggest that velocity constraints and disparity from feature tracking are needed to solve the inverse problem of 3D motion perception. It seems plausible that motion and disparity input is processed in parallel and integrated late in the visual processing hierarchy.     

Solvability of the electrocardiology inverse problem for a moving dipole

New formulations of the direct and inverse problems for the moving dipole are offered. It has been suggested to limit the study by a small area on the chest surface. This lowers the role of the medium inhomogeneity. When formulating the direct problem, irregular components are considered. The algorithm of simultaneous determination of the dipole and regular noise parameters has been described and analytically investigated. It is shown that temporal overdetermination of the equations offers a single solution of the inverse problem for the four leads.     

Estimation of surface heat flux and temperature distributions in a multilayer tissue based on the hyperbolic model of heat conduction

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to solve the inverse hyperbolic heat conduction problem in estimating the unknown time-dependent surface heat flux in a skin tissue, which is stratified into epidermis, dermis, and subcutaneous layers, from the temperature measurements taken within the medium. Subsequently, the temperature distributions in the tissue can be calculated as well. The concept of finite heat propagation velocity is applied to the modeling of the bioheat transfer problem. The inverse solutions will be justified based on the numerical experiments in which two different heat flux distributions are to be determined. The temperature data obtained from the direct problem are used to simulate the temperature measurements. The influence of measurement errors on the precision of the estimated results is also investigated. Results show that an excellent estimation on the time-dependent surface heat flux can be obtained for the test cases considered in this study.     

Image reconstruction of fluorescent molecular tomography based on the tree structured Schur complement decomposition

Background  

The inverse problem of fluorescent molecular tomography (FMT) often involves complex large-scale matrix operations, which may lead to unacceptable computational errors and complexity. In this research, a tree structured Schur complement decomposition strategy is proposed to accelerate the reconstruction process and reduce the computational complexity. Additionally, an adaptive regularization scheme is developed to improve the ill-posedness of the inverse problem.     

Dynamics of granzyme B-induced apoptosis: mathematical modeling

Apoptosis is mediated by an intracellular biochemical system that mainly includes proteins (procaspases, caspases, inhibitors, Bcl-2 protein family as well as substances released from mitochondrial intermembrane space). The dynamics of caspase activation and target cleavage in apoptosis induced by granzyme B in a single K562 cell was studied using a mathematical model of the dynamics of granzyme B-induced apoptosis developed in this work. Also the first application of optimization approach to determination of unknown kinetic constants of biochemical apoptotic reactions was presented. The optimization approach involves solving of two problems: direct and inverse. Solving the direct optimization problem, we obtain the initial (baseline) concentrations of procaspases for known kinetic constants through conditional minimization of a cost function based on the principle of minimum protein consumption by the apoptosis system. The inverse optimization problem is aimed at determination of unknown kinetic constants of apoptotic biochemical reactions proceeding from the condition that the optimal concentrations of procaspases resulting from the solution of the direct optimization problem coincide with the observed ones, that is, those determined by biochemical methods. The Multidimensional Index Method was used to perform numerical solution of the inverse optimization problem.     

Analytical and numerical analysis of inverse optimization problems: conditions of uniqueness and computational methods

One of the key problems of motor control is the redundancy problem, in particular how the central nervous system (CNS) chooses an action out of infinitely many possible. A promising way to address this question is to assume that the choice is made based on optimization of a certain cost function. A number of cost functions have been proposed in the literature to explain performance in different motor tasks: from force sharing in grasping to path planning in walking. However, the problem of uniqueness of the cost function(s) was not addressed until recently. In this article, we analyze two methods of finding additive cost functions in inverse optimization problems with linear constraints, so-called linear-additive inverse optimization problems. These methods are based on the Uniqueness Theorem for inverse optimization problems that we proved recently (Terekhov et?al., J Math Biol 61(3):423?C453, 2010). Using synthetic data, we show that both methods allow for determining the cost function. We analyze the influence of noise on the both methods. Finally, we show how a violation of the conditions of the Uniqueness Theorem may lead to incorrect solutions of the inverse optimization problem.     

An improved inverse dynamics formulation for estimation of external and internal loads during human sagittal plane movements

Planar musculoskeletal models are common in the inverse dynamics analysis of human movements such as walking, running and jumping. The continued interest in such models is justified by their simplicity and computational efficiency. Related to a human planar model, a unified formulation for both the flying and support phases of the sagittal plane movements is developed. The actuation involves muscle forces in the lower limbs and the resultant muscle torques in the other body joints. The dynamic equations, introduced in absolute coordinates of the segments, are converted into useful compact forms using the projective technique. The solution to a determinate inverse dynamics problem allows for the explicit determination of the external reactions (presumed to vanish during the flying phases) and the resultant muscle torques in all the model joints. The indeterminate inverse dynamics problem is then focused on the assessment of muscle forces and joint reaction forces selectively in the supporting lower limb. Numerical results of the inverse dynamics simulation of sample sagittal plane movements are reported to illustrate the validity and effectiveness of the improved formulation.     

Analytical and computational techniques for exogenous depolymerization of xenobiotic polymers

We analyze a mathematical model for exogenous depolymerization of xenobiotic polymers. We derive a sufficient condition for the solvability of its inverse problem to determine degradation rates. We also introduce a numerical technique to solve the inverse problem with an example to show how our numerical technique is applied to gel permeation chromatography data of polyethylene wax obtained before and after three-week cultivation of a fungus Aspergillus sp. AK-3. In addition, we present a numerical result which we obtained by applying the degradation rate based on the three week cultivation to simulate the transition of the weight distribution after cultivation of the fungus for five weeks.     

Inverse sampling of controls in a matched case control study

A method of inverse sampling of controls in a matched case-control study is described in which, for each case, controls are sampled until a discordant set is achieved. For a binary exposure, inverse sampling is used to determine the number of controls for each case. When most individuals in a population have the same exposure, standard case-control sampling may result in many case-control sets being concordant with respect to exposure and thus uninformative in the conditional logistic analysis. The method using inverse control sampling is proposed as a solution to this problem in situations when it is practically feasible. In many circumstances, inverse control sampling is found to offer improved statistical efficiency relative to a comparable study with a fixed number of controls per case.