Alexandre Sedoglavic

A Probabilistic Algorithm to Test Local Algebraic Observability in Polynomial Time

The following questions are often encountered in system and control theory. Given an algebraic model of a physical process, which variables can be, in theory, deduced from the input?output behaviour of an experiment? How many of the remaining variables should we assume to be known in order to determine all the others? These questions are parts of thelocal algebraic observability problem which is concerned with the existence of a non-trivial Lie subalgebra of model?s symmetries letting the inputs and the outputs be invariant.We present a probabilistic seminumerical algorithm that proposes a solution to this problem in polynomial time. A bound for the necessary number of arithmetic operations on the rational field is presented. This bound is polynomial in the complexity of evaluation of the model and in the number of variables. Furthermore, we show that the size of the integers involved in the computations is polynomial in the number of variables and in the degree of the system. Last, we estimate the probability of success of our algorithm.

Aminoglycoside Nephrotoxicity: Modeling, Simulation, and Control

ABSTRACT The main constraints on the administration of aminoglycosides are the risks of nephrotoxicity and ototoxicity, which can lead to acute, renal, vestibular, and auditory toxicities. In the present study we focused on nephrotoxicity. No reliable predictor of nephrotoxicity has been found to date. We have developed a deterministic model which describes the pharmacokinetic behavior of aminoglycosides (with a two-compartment model), the kinetics of aminoglycoside accumulation in the renal cortex, the effects of aminoglycosides on renal cells, the resulting effects on renal function by tubuloglomerular feedback, and the resulting effects on serum creatinine concentrations. The pharmacokinetic parameter values were estimated by use of the NPEM program. The estimated pharmacodynamic parameter values were obtained after minimization of the least-squares objective function between the measured and the calculated serum creatinine concentrations. A simulation program assessed the influences of the dosage regimens on the occurrence of nephrotoxicity. We have also demonstrated the relevancy of modeling of the circadian rhythm of the renal function. We have shown the ability of the model to fit with 49 observed serum creatinine concentrations for a group of eight patients treated for endocarditis by comparison with 49 calculated serum creatinine concentrations (r2 = 0.988; P < 0.001). We have found that for the same daily dose, the nephrotoxicity observed with a thrice-daily administration schedule appears more rapidly, induces a greater decrease in renal function, and is more prolonged than those that occur with less frequent administration schedules (for example, once-daily administration). Moreover, for once-daily administration, we have demonstrated that the time of day of administration can influence the incidence of aminoglycoside nephrotoxicity. The lowest level of nephrotoxicity was observed when aminoglycosides were administered at 1:30 p.m. Clinical application of this model might make it possible to adjust aminoglycoside dosage regimens by taking into account both the efficacies and toxicities of the drugs.

A generalization of flatness to nonlinear systems of partial differential equations. Application to the command of a flexible rod

We introduce a concept of differential flatness for systems described by nonlinear partial differential equations. It generalizes the now classical notion of differential flatness for finite differential systems and its recent extensions to linear partial differential equations. We apply it to the motion planning of a very flexible rod, out of the linear approximation range.

A probabilistic algorithm to test local algebraic observability in polynomial time

The following questions are often encountered in system and control theory. Given an algebraic model of a physical process, which variables can be, in theory, deduced from the input-output behavior of an experiment? How many of the remaining variables should we assume to be known in order to determine all the others? These questions are parts of the local algebraic observability problem which is concerned with the existence of a non trivial Lie subalgebra of the symmetries of the model letting the inputs and the outputs invariant. We present a probabilistic seminumerical algorithm that proposes a solution to this problem in polynomial time. A bound for the necessary number of arithmetic operations on the rational field is presented. This bound is polynomial in the complexity of evaluation of the model and in the number of variables. Furthermore, we show that the size of the integers involved in the computations is polynomial in the number of variables and in the degree of the system. Last, we estimate the probability of success of our algorithm.

Reduction of algebraic parametric systems by rectification of their affine expanded lie symmetries

Lie group theory states that knowledge of a m-parameters solvable group of symmetries of a system of ordinary differential equations allows to reduce by m the number of equations. We apply this principle by finding some affine derivations that induces expanded Lie point symmetries of considered system. By rewriting original problem in an invariant coordinates set for these symmetries, we reduce the number of involved parameters. We present an algorithm based on this standpoint whose arithmetic complexity is quasi-polynomial in inputs size.

Towards an Automated Reduction Method for Polynomial ODE Models of Biochemical Reaction Systems

Abstract.This paper presents the first version of an algorithmic scheme dedicated to the model reduction problem, in the context of polynomial ODE models derived from generalized chemical reaction systems. This scheme, which relies on computer algebra, is implemented within a new MAPLE package. It is applied over an example. The qualitative analysis of the reduced model is afterwards completely carried out, proving the practical relevance of our methods.

A geometric index reduction method for implicit systems of differential algebraic equations

This paper deals with the index reduction problem for the class of quasi-regular DAE systems. It is shown that any of these systems can be transformed to a generically equivalent first order DAE system consisting of a single purely algebraic (polynomial) equation plus an under-determined ODE (a differential Kronecker representation) in as many variables as the order of the input system. This can be done by means of a Kronecker-type algorithm with bounded complexity.

Fast computation of discrete invariants associated to a differential rational mapping

We exhibit probabilistic algorithms which compute the differentiation index, the differential Hilbert function and an algebraic parametric set associated to a differential rational mapping. These algorithms are based on a process of linearization and specialization in a genetic solution, and have polynomial time complexity.

Chemical reaction systems, computer algebra and systems biology

In this invited paper, we survey some of the results obtained in the computer algebra team of Lille, in the domain of systems biology. So far, we have mostly focused on models (systems of equations) arising from generalized chemical reaction systems. Eight years ago, our team was involved in a joint project, with physicists and biologists, on the modeling problem of the circadian clock of the green algae Ostreococcus tauri. This cooperation led us to different algorithms dedicated to the reduction problem of the deterministic models of chemical reaction systems. More recently, we have been working more tightly with another team of our lab, the BioComputing group, interested by the stochastic dynamics of chemical reaction systems. This cooperation led us to efficient algorithms for building the ODE systems which define the statistical moments associated to these dynamics. Most of these algorithms were implemented in the MAPLE computer algebra software. We have chosen to present them through the corresponding MAPLE packages.

The differential Hilbert function of a differential rational mapping can be computed in polynomial time

We present a probabilistic seminumerical algorithm that computes the differential Hilbert function associated to a differential rational mapping. This algorithm explicitly determines the set of variables and derivatives which can be arbitrarily fixed in order to locally invert the differential mapping under consideration. The arithmetic complexity of this algorithm is polynomial in the input size.