340 research outputs found
On identifiability of nonlinear ODE models and applications in viral dynamics
Ordinary differential equations (ODEs) are a powerful tool for modeling dynamic processes
with wide applications in a variety of scientific fields. Over the last two decades, ODEs
have also emerged as a prevailing tool in various biomedical research fields, especially
in infectious disease modeling. In practice, it is important and necessary to determine
unknown parameters in ODE models based on experimental data. Identifiability analysis
is the first step in determining unknown parameters in ODE models and such analysis
techniques for nonlinear ODE models are still under development. In this article, we
review identifiability analysis methodologies for nonlinear ODE models developed in the
past couple of decades, including structural identifiability analysis, practical identifiability
analysis, and sensitivity-based identifiability analysis. Some advanced topics and ongoing
research are also briefly reviewed. Finally, some examples from modeling viral dynamics of
HIV and influenza viruses are given to illustrate how to apply these identifiability analysis
methods in practice.NIAID/NIH
research grants AI055290, AI50020, AI28433, AI078498, RR06555, the University of Rochester
Provost Award, and the University of Rochester DCFAR (P30AI078498) Mentoring Award.http://www.siam.org/journals/sirev/53-1/75700.htmlai201
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
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 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
General analytical condition to nonlinear identifiability and its application in viral dynamics
Identifiability describes the possibility of determining the values of the
unknown parameters that characterize a dynamic system from the knowledge of its
inputs and outputs. This paper finds the general analytical condition that
fully characterizes this property. The condition can be applied to any system,
regardless of its complexity and type of nonlinearity. In the presence of time
varying parameters, it is only required that their time dependence be
analytical. In addition, its implementation requires no inventiveness from the
user as it simply needs to follow the steps of a systematic procedure that only
requires to perform the calculation of derivatives and matrix ranks. Time
varying parameters are treated as unknown inputs and their identifiability is
based on the very recent analytical solution of the unknown input observability
problem. Finally, when a parameter is unidentifiable, the paper also provides
an analytical method to determine infinitely many values for this parameter
that are indistinguishable from its true value. The condition is used to study
the identifiability of two nonlinear models in the field of viral dynamics (HIV
and Covid-19). In particular, regarding the former, a very popular HIV ODE
model is investigated, and the condition allows us to automatically find a new
fundamental result that highlights a serious error in the current state of the
art.Comment: This preprint is currently under review on Transaction and Automatic
Control. It is a short version of arXiv:2211.13507. It includes the
definition of identifiability in the presence of time varying parameters -
which is absent in arXiv:2211.1350
Dynamical compensation and structural identifiability: analysis, implications, and reconciliation
The concept of dynamical compensation has been recently introduced to
describe the ability of a biological system to keep its output dynamics
unchanged in the face of varying parameters. Here we show that, according to
its original definition, dynamical compensation is equivalent to lack of
structural identifiability. This is relevant if model parameters need to be
estimated, which is often the case in biological modelling. This realization
prompts us to warn that care should we taken when using an unidentifiable model
to extract biological insight: the estimated values of structurally
unidentifiable parameters are meaningless, and model predictions about
unmeasured state variables can be wrong. Taking this into account, we explore
alternative definitions of dynamical compensation that do not necessarily imply
structural unidentifiability. Accordingly, we show different ways in which a
model can be made identifiable while exhibiting dynamical compensation. Our
analyses enable the use of the new concept of dynamical compensation in the
context of parameter identification, and reconcile it with the desirable
property of structural identifiability
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