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.     

Structure-approximating inverse protein folding problem in the 2D HP model.

The inverse protein folding problem is that of designing an amino acid sequence which has a particular native protein fold. This problem arises in drug design where a particular structure is necessary to ensure proper protein-protein interactions. In this paper, we show that in the 2D HP model of Dill it is possible to solve this problem for a broad class of structures. These structures can be used to closely approximate any given structure. One of the most important properties of a good protein (in drug design) is its stability--the aptitude not to fold simultaneously into other structures. We show that for a number of basic structures, our sequences have a unique fold.     

A model of ventilatory instability induced in the unrestrained rat.

A classic conditioning paradigm was used to examine the hypothesis that perturbations during sleep in the neonate rat can have a lasting impact on breathing. During the first 4 wk of life, stimuli were presented to rats during behaviorally defined sleep. In a conditioned hypoxic (CH) group, brief periods of hypoxic gas were used as the unconditioned stimulus. Tactile and auditory stimuli were used as the conditioned stimuli. In a conditioned control (CC) group, air was used as the unconditioned stimulus. A third group of unconditioned control (UC) rats was not exposed to the conditioning paradigm. Animals were provided routine care for 3.5 mo; ventilation was then assessed using plethysmography. Conditioning during neonatal life produced increased ventilatory irregularities and apnea during behaviorally defined sleep in adult rats. Both CH and CC rats showed a significantly greater number of apneic events compared with UC rats. Over a 2-h sleep period, CH rats exhibited a total of 105.1 +/- 9.4 (SE) apneic events, CC rats 69.4 +/- 4.2 events, and UC rats 42.1 +/- 3.1 events [F(2,18) = 25.568; P < 0.0001]. These findings suggest that experiences in the first few weeks of life will alter ventilatory patterning in the adult animal.     

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.     

Comparison and testing of least-squares time domain inverse solutions in electrocardiography

The use of several mathematical methods for estimating epicardial ECG potentials from arrays of body surface potentials has been reported in the literature; most of these methods are based on least-squares reconstruction principles and operate in the time-space domain. In this paper we introduce a general Bayesian maximum a posteriori (MAP) framework for time domain inverse solutions in the presence of noise. The two most popular previously applied least-squares methods, constrained (regularized) least-squares and low-rank approximation through the singular value decomposition, are placed in this framework, each of them requiring the a priori knowledge of a ‘regularization parameter’, which defines the degree of smoothing to be applied to the inversion. Results of simulations using these two methods are presented; they compare the ability of each method to reconstruct epicardial potentials. We used the geometric configuration of the torso and internal organs of an individual subject as reconstructed from CT scans. The accuracy of each method at each epicardial location was tested as a function of measurement noise, the size and shape of the subarray of torso sensors, and the regularization parameter. We paid particular attention to an assessment of the potential of these methods for clinical use by testing the effect of using compact, small-size subarrays of torso potentials while maintaining a high degree of resolution on the epicardium.     

Topology fingerprint approach to the inverse protein folding problem.

We describe the most general solution to date of the problem of matching globular protein sequences to the appropriate three-dimensional structures. The screening template, against which sequences are tested, is provided by a protein "structural fingerprint" library based on the contact map and the buried/exposed pattern of residues. Then, a lattice Monte Carlo algorithm validates or dismisses the stability of the proposed fold. Examples of known structural similarities between proteins having weakly or unrelated sequences such as the globins and phycocyanins, the eight-member alpha/beta fold of triose phosphate isomerase and even a close structural equivalence between azurin and immunoglobulins are found.     

A two-dimension otolith growth inverse model

An inverse growth model was developed to describe fish otolith ontogenetic growth to find the most parsimonious growth field that drives the actual shape of an internal incremental structure to the actual external shape of the otolith. This approach is based in a pure physic model, and the single implicit assumption was that the otolith accretion occurred perpendicular to its external surface. The model could be used to define a general hypothesis situation, in the sense that it predicts the most plausible shape of the otolith at any time, assuming that no additional external forces have acted. Therefore, severe departures between the observed growth structure and the closest prediction to it could be used to identify anatomical constraints and physiological factors that modulate the pure physical expectations.     

An inverse problem in neural processing

Any neural network aimed at the coding sensory events must contain computational properties which generally allow the organism to reconstruct the input signals with some degree of accuracy-else the association between stimulus and response would, at best, be uncertain. In this paper we investigate the problem of reconstructing external input signals to neural networks when the activity profiles of only some of its member cells are known. The evolution and activities of such cells are defined by an earlier formulation of one of us (Ouztöreli 1979) and, here, we restrict our application to local curcuits within the vertebrate retina. Solutions to this inverse coding problem are presented for specific network equations and examplified with 1, 3, and 5 neuron cases.This work was partially supported by the Natural Sciences and Engineering Research Council of Canada under Grant A-4345 to M.N.O. and grant A-4395 to T.M.C. through the University of Alberta     

Hyperthermophiles and the problem of DNA instability

Rates of chemical decomposition of DNA at the optimal growth temperatures of hyperthermophiles seem incongruent with the requirements of accurate genome replication. The peculiar physiology, ecology and phylogeny of hyperthermophiles combine to suggest that these prokaryotes have solved a molecular problem (spontaneous loss of native DNA structure) of a magnitude that well-studied microorganisms do not face. The failure of DNA base composition to correlate with optimal growth temperature among hyperthermophiles provides indirect evidence that other mechanisms maintain their chromosomal DNA in the duplex form. Studies in vitro indicate that DNA primary structure is more difficult to maintain at extremely high temperature than is secondary structure, yet hyperthermophiles exhibit only modest levels of spontaneous mutation. Radiation sensitivity studies also indicate that hyperthermophiles repair their DNA efficiently in vivo , and underlying mechanisms are beginning to be examined. Several enzymes of DNA metabolism from hyperthermophilic archaea exhibit unusual biochemical features that may ultimately prove relevant to DNA repair. However, genomic sequencing results suggest that many DNA repair genes of hyperthermophilic archaea may not be recognized because they are not sufficiently related to those of well-studied organisms.     

The fundamental problem of myoskeletal inverse dynamics and its implications.

The validity of current inverse dynamics models utilized for motion analysis is investigated. It is shown that observables generated by the real biosystem, such as ground reaction forces, are incompatible with comparable responses of skeletodynamical inverse models currently in use. This implies that results obtained with such models are erroneous to varying degrees while a quantification of these errors is difficult or impossible. This phenomenon is termed the fundamental myoskeletal inverse dynamics problem. A model fidelity indicator is proposed which, for a specific inverse dynamics model applied to a particular motion, provides a dimensionless numerical measure for the replicative validity of that model and the fidelity of its input data. A practical example demonstrates the usefulness of this indicator. It is suggested that the development of structurally sufficiently complex and biologically more realistic skeletomechanical models as well as substantial error reductions in data measuring and processing procedures will be necessary to improve the accuracy of inverse dynamics model computations.     

Ambiguity of the solution of the inverse problem of electroencephalography

It has been shown for homogeneous conducting medium that if different stationary distributions of current sources form identical potential fields on the plane boundary, the difference between these two distributions is the distribution of current sources with all multiple moments equaling zero. Examples of current source distributions are presented which satisfy this condition.     

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.     

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     

Molecular seismology: an inverse problem in nanobiology

The density profile of an elastic fiber like DNA will change in space and time as ligands associate with it. This observation affords a new direction in single molecule studies provided that density profiles can be measured in space and time. In fact, this is precisely the objective of seismology, where the mathematics of inverse problems have been employed with success. We argue that inverse problems in elastic media can be directly applied to biophysical problems of fiber-ligand association, and demonstrate that robust algorithms exist to perform density reconstruction in the condensed phase.     

An inverse problem in estimating the laser irradiance and thermal damage in laser-irradiated biological tissue with a dual-phase-lag model

The aim of this study is to solve an inverse heat conduction problem to estimate the unknown time-dependent laser irradiance and thermal damage in laser-irradiated biological tissue from the temperature measurements taken within the tissue. The dual-phase-lag model is considered in the formulation of heat conduction equation. The inverse algorithm used in the study is based on the conjugate gradient method and the discrepancy principle. The effect of measurement errors and measurement locations on the estimation accuracy is also investigated. Two different examples of laser irradiance are discussed. Results show that the unknown laser irradiance and thermal damage can be predicted precisely by using the present approach for the test cases considered in this study.     

A genetic algorithm approach to the inverse problem of treatment planning for intensity-modulated radiotherapy

A genetic algorithm optimization approach for designing treatment plans in intensity-modulated radiotherapy is proposed. The approach determines the beam intensities of the pencil-beam dose model such that the optimized dose distribution closely matches the prescribed dose distribution. The approach indirectly inverts the ill-conditioned dose-projection matrix, which can be very large and extremely sparse. The beam intensities are treated as chromosomes that are encoded as binary strings. The approach was used to design treatment plans for two deceptive clinical test cases. In both case, cancerous tissues in the planning target region received at least 98% of the prescribed dose level while dose levels delivered to the organs at risk were well within safe limits, with a maximum exposure of 2.5 and 52.5% of the prescribed tolerance level for the brain and prostrate cancer cases, respectively. Dose levels delivered to the healthy tissues were small with a mean exposure of 22.8 and 23.5% of the prescribed tolerance level.     

Adaptive local regularization methods for the inverse ECG problem

One of the fundamental problems in theoretical electrocardiography can be characterized by an inverse problem. We present new methods for achieving better estimates of heart surface potential distributions in terms of torso potentials through an inverse procedure. First, we outline an automatic adaptive refinement algorithm that minimizes the spatial discretization error in the transfer matrix, increasing the accuracy of the inverse solution. Second, we introduce a new local regularization procedure, which works by partitioning the global transfer matrix into sub-matrices, allowing for varying amounts of smoothing. Each submatrix represents a region within the underlying geometric model in which regularization can be specifically ‘tuned’ using an a priori scheme based on the L-curve method. This local regularization method can provide a substantial increase in accuracy compared to global regularization schemes. Within this context of local regularization, we show that a generalized version of the singular value decomposition (GSVD) can further improve the accuracy of ECG inverse solutions compared to standard SVD and Tikhonov approaches. We conclude with specific examples of these techniques using geometric models of the human thorax derived from MRI data.