353 research outputs found
Estimation of constant and time-varying dynamic parameters of HIV infection in a nonlinear differential equation model
Modeling viral dynamics in HIV/AIDS studies has resulted in a deep
understanding of pathogenesis of HIV infection from which novel antiviral
treatment guidance and strategies have been derived. Viral dynamics models
based on nonlinear differential equations have been proposed and well developed
over the past few decades. However, it is quite challenging to use experimental
or clinical data to estimate the unknown parameters (both constant and
time-varying parameters) in complex nonlinear differential equation models.
Therefore, investigators usually fix some parameter values, from the literature
or by experience, to obtain only parameter estimates of interest from clinical
or experimental data. However, when such prior information is not available, it
is desirable to determine all the parameter estimates from data. In this paper
we intend to combine the newly developed approaches, a multi-stage
smoothing-based (MSSB) method and the spline-enhanced nonlinear least squares
(SNLS) approach, to estimate all HIV viral dynamic parameters in a nonlinear
differential equation model. In particular, to the best of our knowledge, this
is the first attempt to propose a comparatively thorough procedure, accounting
for both efficiency and accuracy, to rigorously estimate all key kinetic
parameters in a nonlinear differential equation model of HIV dynamics from
clinical data. These parameters include the proliferation rate and death rate
of uninfected HIV-targeted cells, the average number of virions produced by an
infected cell, and the infection rate which is related to the antiviral
treatment effect and is time-varying. To validate the estimation methods, we
verified the identifiability of the HIV viral dynamic model and performed
simulation studies.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS290 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Sieve estimation of constant and time-varying coefficients in nonlinear ordinary differential equation models by considering both numerical error and measurement error
This article considers estimation of constant and time-varying coefficients
in nonlinear ordinary differential equation (ODE) models where analytic
closed-form solutions are not available. The numerical solution-based nonlinear
least squares (NLS) estimator is investigated in this study. A numerical
algorithm such as the Runge--Kutta method is used to approximate the ODE
solution. The asymptotic properties are established for the proposed estimators
considering both numerical error and measurement error. The B-spline is used to
approximate the time-varying coefficients, and the corresponding asymptotic
theories in this case are investigated under the framework of the sieve
approach. Our results show that if the maximum step size of the -order
numerical algorithm goes to zero at a rate faster than , the
numerical error is negligible compared to the measurement error. This result
provides a theoretical guidance in selection of the step size for numerical
evaluations of ODEs. Moreover, we have shown that the numerical solution-based
NLS estimator and the sieve NLS estimator are strongly consistent. The sieve
estimator of constant parameters is asymptotically normal with the same
asymptotic co-variance as that of the case where the true ODE solution is
exactly known, while the estimator of the time-varying parameter has the
optimal convergence rate under some regularity conditions. The theoretical
results are also developed for the case when the step size of the ODE numerical
solver does not go to zero fast enough or the numerical error is comparable to
the measurement error. We illustrate our approach with both simulation studies
and clinical data on HIV viral dynamics.Comment: Published in at http://dx.doi.org/10.1214/09-AOS784 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Optimal Rate of Direct Estimators in Systems of Ordinary Differential Equations Linear in Functions of the Parameters
Many processes in biology, chemistry, physics, medicine, and engineering are
modeled by a system of differential equations. Such a system is usually
characterized via unknown parameters and estimating their 'true' value is thus
required. In this paper we focus on the quite common systems for which the
derivatives of the states may be written as sums of products of a function of
the states and a function of the parameters.
For such a system linear in functions of the unknown parameters we present a
necessary and sufficient condition for identifiability of the parameters. We
develop an estimation approach that bypasses the heavy computational burden of
numerical integration and avoids the estimation of system states derivatives,
drawbacks from which many classic estimation methods suffer. We also suggest an
experimental design for which smoothing can be circumvented. The optimal rate
of the proposed estimators, i.e., their -consistency, is proved and
simulation results illustrate their excellent finite sample performance and
compare it to other estimation approaches
A dynamic Bayesian nonlinear mixed-effects model of HIV response incorporating medication adherence, drug resistance and covariates
HIV dynamic studies have contributed significantly to the understanding of
HIV pathogenesis and antiviral treatment strategies for AIDS patients.
Establishing the relationship of virologic responses with clinical factors and
covariates during long-term antiretroviral (ARV) therapy is important to the
development of effective treatments. Medication adherence is an important
predictor of the effectiveness of ARV treatment, but an appropriate determinant
of adherence rate based on medication event monitoring system (MEMS) data is
critical to predict virologic outcomes. The primary objective of this paper is
to investigate the effects of a number of summary determinants of MEMS
adherence rates on virologic response measured repeatedly over time in
HIV-infected patients. We developed a mechanism-based differential equation
model with consideration of drug adherence, interacted by virus susceptibility
to drug and baseline characteristics, to characterize the long-term virologic
responses after initiation of therapy. This model fully integrates viral load,
MEMS adherence, drug resistance and baseline covariates into the data analysis.
In this study we employed the proposed model and associated Bayesian nonlinear
mixed-effects modeling approach to assess how to efficiently use the MEMS
adherence data for prediction of virologic response, and to evaluate the
predicting power of each summary metric of the MEMS adherence rates.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS376 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Observability and Structural Identifiability of Nonlinear Biological Systems
Observability is a modelling property that describes the possibility of
inferring the internal state of a system from observations of its output. A
related property, structural identifiability, refers to the theoretical
possibility of determining the parameter values from the output. In fact,
structural identifiability becomes a particular case of observability if the
parameters are considered as constant state variables. It is possible to
simultaneously analyse the observability and structural identifiability of a
model using the conceptual tools of differential geometry. Many complex
biological processes can be described by systems of nonlinear ordinary
differential equations, and can therefore be analysed with this approach. The
purpose of this review article is threefold: (I) to serve as a tutorial on
observability and structural identifiability of nonlinear systems, using the
differential geometry approach for their analysis; (II) to review recent
advances in the field; and (III) to identify open problems and suggest new
avenues for research in this area.Comment: Accepted for publication in the special issue "Computational Methods
for Identification and Modelling of Complex Biological Systems" of Complexit
A Sensitivity Matrix Methodology for Inverse Problem Formulation
We propose an algorithm to select parameter subset combinations that can be estimated using an ordinary least-squares (OLS) inverse problem formulation with a given data set. First, the algorithm selects the parameter combinations that correspond to sensitivity matrices with full rank. Second, the algorithm involves uncertainty quantification by using the inverse of the Fisher Information Matrix. Nominal values of parameters are used to construct synthetic data sets, and explore the effects of removing certain parameters from those to be estimated using OLS procedures. We quantify these effects in a score for a vector parameter defined using the norm of the vector of standard errors for components of estimates divided by the estimates. In some cases the method leads to reduction of the standard error for a parameter to less than 1% of the estimate
Observability, Identifiability and Epidemiology -- A survey
In this document we introduce the concepts of Observability and
Iden-tifiability in Mathematical Epidemiology. We show that, even for simple
and well known models, these properties are not always fulfilled. We also
consider the problem of practical observability and identi-fiability which are
connected to sensitivity and numerical condition numbers
Modeling the dynamics of biomarkers during primary HIV infection taking into account the uncertainty of infection date
During primary HIV infection, the kinetics of plasma virus concentrations and
CD4+ cell counts is very complex. Parametric and nonparametric models have been
suggested for fitting repeated measurements of these markers. Alternatively,
mechanistic approaches based on ordinary differential equations have also been
proposed. These latter models are constructed according to biological knowledge
and take into account the complex nonlinear interactions between viruses and
cells. However, estimating the parameters of these models is difficult. A main
difficulty in the context of primary HIV infection is that the date of
infection is generally unknown. For some patients, the date of last negative
HIV test is available in addition to the date of first positive HIV test
(seroconverters). In this paper we propose a likelihood-based method for
estimating the parameters of dynamical models using a population approach and
taking into account the uncertainty of the infection date. We applied this
method to a sample of 761 HIV-infected patients from the Concerted Action on
SeroConversion to AIDS and Death in Europe (CASCADE).Comment: Published in at http://dx.doi.org/10.1214/10-AOAS364 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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