Modeling and Estimation of Kinetic Parameters and Replicative Fitness of HIV-1 from Flow-Cytometry-Based Growth Competition Experiments

Growth competition assays have been developed to quantify the relative fitness of HIV-1 mutants. In this article, we develop mathematical models to describe viral/cellular dynamic interactions in the assay system from which the competitive fitness indices or parameters are defined. In our previous HIV-viral fitness experiments, the concentration of uninfected target cells was assumed to be constant (Wu et al. 2006). But this may not be true in some experiments. In addition, dual infection may frequently occur in viral fitness experiments and may not be ignorable. Here, we relax these two assumptions and extend our earlier viral fitness model (Wu et al. 2006). The resulting models then become nonlinear ODE systems for which closed-form solutions are not achievable. In the new model, the viral relative fitness is a function of time since it depends on the target cell concentration. First, we studied the structure identifiability of the nonlinear ODE models. The identifiability analysis showed that all parameters in the proposed models are identifiable from the flow-cytometry-based experimental data that we collected. We then employed a global optimization approach (the differential evolution algorithm) to directly estimate the kinetic parameters as well as the relative fitness index in the nonlinear ODE models using nonlinear least square regression based on the experimental data. Practical identifiability was investigated via Monte Carlo simulations.     

Hierarchical Bayesian methods for estimation of parameters in a longitudinal HIV dynamic system

HIV dynamics studies have significantly contributed to the understanding of HIV infection and antiviral treatment strategies. But most studies are limited to short-term viral dynamics due to the difficulty of establishing a relationship of antiviral response with multiple treatment factors such as drug exposure and drug susceptibility during long-term treatment. In this article, a mechanism-based dynamic model is proposed for characterizing long-term viral dynamics with antiretroviral therapy, described by a set of nonlinear differential equations without closed-form solutions. In this model we directly incorporate drug concentration, adherence, and drug susceptibility into a function of treatment efficacy, defined as an inhibition rate of virus replication. We investigate a Bayesian approach under the framework of hierarchical Bayesian (mixed-effects) models for estimating unknown dynamic parameters. In particular, interest focuses on estimating individual dynamic parameters. The proposed methods not only help to alleviate the difficulty in parameter identifiability, but also flexibly deal with sparse and unbalanced longitudinal data from individual subjects. For illustration purposes, we present one simulation example to implement the proposed approach and apply the methodology to a data set from an AIDS clinical trial. The basic concept of the longitudinal HIV dynamic systems and the proposed methodologies are generally applicable to any other biomedical dynamic systems.     

Construction of a large scale integrated map of macrophage pathogen recognition and effector systems

Background

Mathematical models provide abstract representations of the information gained from experimental observations on the structure and function of a particular biological system. Conferring a predictive character on a given mathematical formulation often relies on determining a number of non-measurable parameters that largely condition the model's response. These parameters can be identified by fitting the model to experimental data. However, this fit can only be accomplished when identifiability can be guaranteed.

Results

We propose a novel iterative identification procedure for detecting and dealing with the lack of identifiability. The procedure involves the following steps: 1) performing a structural identifiability analysis to detect identifiable parameters; 2) globally ranking the parameters to assist in the selection of the most relevant parameters; 3) calibrating the model using global optimization methods; 4) conducting a practical identifiability analysis consisting of two (a priori and a posteriori) phases aimed at evaluating the quality of given experimental designs and of the parameter estimates, respectively and 5) optimal experimental design so as to compute the scheme of experiments that maximizes the quality and quantity of information for fitting the model.

Conclusions

The presented procedure was used to iteratively identify a mathematical model that describes the NF-κB regulatory module involving several unknown parameters. We demonstrated the lack of identifiability of the model under typical experimental conditions and computed optimal dynamic experiments that largely improved identifiability properties.     

On transient effects in the HIV/AIDS epidemic

During the initially exponential spread of the human immunodeficiency virus (HIV—the causative agent of AIDS) the growth rate of the number of AIDS cases decreases from plus infinity to the growth rate of HIV infections. A sensitivity analysis shows that for all reasonable values of the parameters of the HIV epidemic (incubation period, initial doubling time, etc.) the effect of this positive transient becomes negligible when the annual number of AIDS cases reaches a few dozen. Necessary and sufficient conditions are given for the growth rate of the number of AIDS cases to be monotonically decreasing during the positive transient. A mildly pathological density function for the incubation period of AIDS provides an example of a growth rate of AIDS that does not decrease monotonically, even though HIV is spreading exponentially. A negative transient occurs when the growth rate of HIV begins to decrease. In this context a somewhat surprising result emerges under the assumption that the growth rate of HIV is non-increasing: the growth rate of AIDS is at all times larger than the growth rate of HIV. A logistic HIV epidemic illustrates this result, and implications for the growth of the HIV epidemic in the United States and Europe are discussed. In particular, it is shown that the positive transient must have passed by 1982 in the United States and by 1986 or 1987 for the five European countries with the largest caseloads.     

Practical Identifiability of HIV Dynamics Models

We study the practical identifiability of parameters, i.e., the accuracy of the estimation that can be hoped, in a model of HIV dynamics based on a system of non-linear Ordinary Differential Equations (ODE). This depends on the available information such as the schedule of the measurements, the observed components, and the measurement precision. The number of patients is another way to increase it by introducing an appropriate statistical “population” framework. The impact of each improvement of the experimental condition is not known in advance but it can be evaluated via the Fisher Information Matrix (FIM). If the non-linearity of the biological model, as well as the complex statistical framework makes computation of the FIM challenging, we show that the particular structure of these models enables to compute it as precisely as wanted. In the HIV model, measuring HIV viral load and total CD4+ count were not enough to achieve identifiability of all the parameters involved. However, we show that an appropriate statistical approach together with the availability of additional markers such as infected cells or activated cells should considerably improve the identifiability and thus the usefulness of dynamical models of HIV.     

The Innate Immune Response in HIV/AIDS Septic Shock Patients: A Comparative Study

Introduction

In recent years, the incidence of sepsis has increased in critically ill HIV/AIDS patients, and the presence of severe sepsis emerged as a major determinant of outcomes in this population. The inflammatory response and deregulated cytokine production play key roles in the pathophysiology of sepsis; however, these mechanisms have not been fully characterized in HIV/AIDS septic patients.

Methods

We conducted a prospective cohort study that included HIV/AIDS and non-HIV patients with septic shock. We measured clinical parameters and biomarkers (C-reactive protein and cytokine levels) on the first day of septic shock and compared these parameters between HIV/AIDS and non-HIV patients.

Results

We included 30 HIV/AIDS septic shock patients and 30 non-HIV septic shock patients. The HIV/AIDS patients presented low CD4 cell counts (72 [7-268] cells/mm³), and 17 (57%) patients were on HAART before hospital admission. Both groups were similar according to the acute severity scores and hospital mortality. The IL-6, IL-10 and G-CSF levels were associated with hospital mortality in the HIV/AIDS septic group; however, the CRP levels and the surrogates of innate immune activation (cytokines) were similar among HIV/AIDS and non-HIV septic patients. Age (odds ratio 1.05, CI 95% 1.02-1.09, p=0.002) and the IL-6 levels (odds ratio 1.00, CI 95% 1.00-1.01, p=0.05) were independent risk factors for hospital mortality.

Conclusions

IL-6, IL-10 and G-CSF are biomarkers that can be used to predict prognosis and outcomes in HIV/AIDS septic patients. Although HIV/AIDS patients are immunocompromised, an innate immune response can be activated in these patients, which is similar to that in the non-HIV septic population. In addition, age and the IL-6 levels are independent risk factors for hospital mortality irrespective of HIV/AIDS disease.     

Practical identifiability of biokinetic parameters of a model describing two-step nitrification in biofilms

Parameter estimation and model calibration are key problems in the application of biofilm models in engineering practice, where a large number of model parameters need to be determined usually based on experimental data with only limited information content. In this article, identifiability of biokinetic parameters of a biofilm model describing two-step nitrification was evaluated based solely on bulk phase measurements of ammonium, nitrite, and nitrate. In addition to evaluating the impact of experimental conditions and available measurements, the influence of mass transport limitation within the biofilm and the initial parameter values on identifiability of biokinetic parameters was evaluated. Selection of parameters for identifiability analysis was based on global mean sensitivities while parameter identifiability was analyzed using local sensitivity functions. At most, four of the six most sensitive biokinetic parameters were identifiable from results of batch experiments at bulk phase dissolved oxygen concentrations of 0.8 or 5 mg O(2)/L. High linear dependences between the parameters of the subsets (KO2,AOB,muAOB) and (KO2,NOB,muNOB) resulted in reduced identifiability. Mass transport limitation within the biofilm did not influence the number of identifiable parameters but, in fact, decreased collinearity between parameters, especially for parameters that are otherwise correlated (e.g., muAOB) and KO2,AOB, or muNOB and KO2,NOB). The choice of the initial parameter values had a significant impact on the identifiability of two parameter subsets, both including the parameters muAOB and KO2,AOB. Parameter subsets that did not include the subsets muAOB and KO2,AOB or muNOB and KO2,NOB were clearly identifiable independently of the choice of the initial parameter values.     

Structural identifiability of polynomial and rational systems

Since analysis and simulation of biological phenomena require the availability of their fully specified models, one needs to be able to estimate unknown parameter values of the models. In this paper we deal with identifiability of parametrizations which is the property of one-to-one correspondence of parameter values and the corresponding outputs of the models. Verification of identifiability of a parametrization precedes estimation of numerical values of parameters, and thus determination of a fully specified model of a considered phenomenon. We derive necessary and sufficient conditions for the parametrizations of polynomial and rational systems to be structurally or globally identifiable. The results are applied to investigate the identifiability properties of the system modeling a chain of two enzyme-catalyzed irreversible reactions. The other examples deal with the phenomena modeled by using Michaelis–Menten kinetics and the model of a peptide chain elongation.     

The use of operational modeling of HIV/AIDS in a systems approach to public health decision making.

Compartmental models of infectious diseases readily represent known biological and epidemiological processes, are easily understood in flow-chart form by administrators, are simple to adjust to new information, and lend themselves to routine statistical analysis such as parameter estimation and model fitting. Technical results are immediately interpretable in epidemiological and public health terms. Deterministic models are easily stochasticized where this is important for practical purposes. With HIV/AIDS, serial data on both HIV prevalence and AIDS morbidity have been available from San Francisco. Assuming the distribution of the incubation period to be biologically stable, statistical analysis is quite feasible in other regions, even those with no reliable HIV data. Transmission rates must be estimated locally. It is also often possible to estimate the effective size of a population subgroup at risk, from population data on AIDS morbidity only. Computer simulation provides estimates of the evolving pattern of both HIV prevalence and AIDS morbidity. Some public health questions can be answered only by appropriately formulated stochastic models.     

HIV/AIDS dynamics: Impact of economic classes with transmission from poor clinical settings

We formulate and analyze a nonlinear deterministic HIV/AIDS model with two social classes, namely the poor and the rich including transmission from poor clinical settings with a randomly variable population. Four sub-models are derived from the full model, the disease threshold parameters are computed, and it is shown that the disease will die down if these initial threshold parameters are less than unity and will persist if they exceed unity. The impact of economic classes (along with transmission from poor/inadequate clinical settings) on the disease dynamics is assessed, and we observe that even with a single sexual partner, the reproduction number is slightly greater than unity, implying that the additional transmission can only be from clinical settings. Stability (local and global) of both the disease-free and endemic equilibria are then investigated using various techniques of dynamical systems such as Centre Manifold theory and Lyapunov's second method. Analysis on the bifurcation parameter is carried out to assess the impact of related HIV transmission from poor clinical settings. We estimate some of the model parameter values and numerical simulations of the model are represented graphically. Our results show that the prevalence of HIV in rich communities seems to be higher than that in the poor, but the disease develops faster in impoverished individuals.     

Structural identifiability of systems biology models: a critical comparison of methods

Analysing the properties of a biological system through in silico experimentation requires a satisfactory mathematical representation of the system including accurate values of the model parameters. Fortunately, modern experimental techniques allow obtaining time-series data of appropriate quality which may then be used to estimate unknown parameters. However, in many cases, a subset of those parameters may not be uniquely estimated, independently of the experimental data available or the numerical techniques used for estimation. This lack of identifiability is related to the structure of the model, i.e. the system dynamics plus the observation function. Despite the interest in knowing a priori whether there is any chance of uniquely estimating all model unknown parameters, the structural identifiability analysis for general non-linear dynamic models is still an open question. There is no method amenable to every model, thus at some point we have to face the selection of one of the possibilities. This work presents a critical comparison of the currently available techniques. To this end, we perform the structural identifiability analysis of a collection of biological models. The results reveal that the generating series approach, in combination with identifiability tableaus, offers the most advantageous compromise among range of applicability, computational complexity and information provided.     

Parameter Identifiability and Redundancy: Theoretical Considerations

Background

Models for complex biological systems may involve a large number of parameters. It may well be that some of these parameters cannot be derived from observed data via regression techniques. Such parameters are said to be unidentifiable, the remaining parameters being identifiable. Closely related to this idea is that of redundancy, that a set of parameters can be expressed in terms of some smaller set. Before data is analysed it is critical to determine which model parameters are identifiable or redundant to avoid ill-defined and poorly convergent regression.

Methodology/Principal Findings

In this paper we outline general considerations on parameter identifiability, and introduce the notion of weak local identifiability and gradient weak local identifiability. These are based on local properties of the likelihood, in particular the rank of the Hessian matrix. We relate these to the notions of parameter identifiability and redundancy previously introduced by Rothenberg (Econometrica 39 (1971) 577–591) and Catchpole and Morgan (Biometrika 84 (1997) 187–196). Within the widely used exponential family, parameter irredundancy, local identifiability, gradient weak local identifiability and weak local identifiability are shown to be largely equivalent. We consider applications to a recently developed class of cancer models of Little and Wright (Math Biosciences 183 (2003) 111–134) and Little et al. (J Theoret Biol 254 (2008) 229–238) that generalize a large number of other recently used quasi-biological cancer models.

Conclusions/Significance

We have shown that the previously developed concepts of parameter local identifiability and redundancy are closely related to the apparently weaker properties of weak local identifiability and gradient weak local identifiability—within the widely used exponential family these concepts largely coincide.     

The relationship between virologic and immunologic responses in AIDS clinical research using mixed-effects varying-coefficient models with measurement error

In this article we study the relationship between virologic and immunologic responses in AIDS clinical trials. Since plasma HIV RNA copies (viral load) and CD4+ cell counts are crucial virologic and immunologic markers for HIV infection, it is important to study their relationship during HIV/AIDS treatment. We propose a mixed-effects varying-coefficient model based on an exploratory analysis of data from a clinical trial. Since both viral load and CD4+ cell counts are subject to measurement error, we also consider the measurement error problem in covariates in our model. The regression spline method is proposed for inference for parameters in the proposed model. The regression spline method transforms the unknown nonparametric components into parametric functions. It is relatively simple to implement using readily available software, and parameter inference can be developed from standard parametric models. We apply the proposed models and methods to an AIDS clinical study. From this study, we find an interesting relationship between viral load and CD4+ cell counts during antiviral treatments. Biological interpretations and clinical implications are discussed.     

Age and Sex Structured Model for Assessing the Demographic Impact of Mother-to-Child Transmission of HIV/AIDS

Age and sex structured HIV/AIDS model with explicit incubation period is proposed as a system of delay differential equations. The model consists of two age groups that are children (0–14 years) and adults (15–49 years). Thus, the model considers both mother-to-child transmission (MTCT) and heterosexual transmission of HIV in a community. MTCT can occur prenatally, at labour and delivery or postnatally through breastfeeding. In the model, we consider the children age group as a one-sex formulation and divide the adult age group into a two-sex structure consisting of females and males. The important mathematical features of the model are analysed. The disease-free and endemic equilibria are found and their stabilities investigated. We use the Lyapunov functional approach to show the local stability of the endemic equilibrium. Qualitative analysis of the model including positivity and boundedness of solutions, and persistence are also presented. The basic reproductive number (ℛ₀) for the model shows that the adult population is responsible for the spread HIV/AIDS epidemic, thus up-to-date developed HIV/AIDS models to assess intervention strategies have focused much on heterosexual transmission by the adult population and the children population has received little attention. We numerically analyse the HIV/AIDS model to assess the community benefits of using antiretroviral drugs in reducing MTCT and the effects of breastfeeding in settings with high HIV/AIDS prevalence ratio using demographic and epidemiological parameters for Zimbabwe.     

On Finding and Using Identifiable Parameter Combinations in Nonlinear Dynamic Systems Biology Models and COMBOS: A Novel Web Implementation

Parameter identifiability problems can plague biomodelers when they reach the quantification stage of development, even for relatively simple models. Structural identifiability (SI) is the primary question, usually understood as knowing which of P unknown biomodel parameters p ₁,…, p_i,…, p_P are-and which are not-quantifiable in principle from particular input-output (I-O) biodata. It is not widely appreciated that the same database also can provide quantitative information about the structurally unidentifiable (not quantifiable) subset, in the form of explicit algebraic relationships among unidentifiable p_i. Importantly, this is a first step toward finding what else is needed to quantify particular unidentifiable parameters of interest from new I–O experiments. We further develop, implement and exemplify novel algorithms that address and solve the SI problem for a practical class of ordinary differential equation (ODE) systems biology models, as a user-friendly and universally-accessible web application (app)–COMBOS. Users provide the structural ODE and output measurement models in one of two standard forms to a remote server via their web browser. COMBOS provides a list of uniquely and non-uniquely SI model parameters, and–importantly-the combinations of parameters not individually SI. If non-uniquely SI, it also provides the maximum number of different solutions, with important practical implications. The behind-the-scenes symbolic differential algebra algorithms are based on computing Gröbner bases of model attributes established after some algebraic transformations, using the computer-algebra system Maxima. COMBOS was developed for facile instructional and research use as well as modeling. We use it in the classroom to illustrate SI analysis; and have simplified complex models of tumor suppressor p53 and hormone regulation, based on explicit computation of parameter combinations. It’s illustrated and validated here for models of moderate complexity, with and without initial conditions. Built-in examples include unidentifiable 2 to 4-compartment and HIV dynamics models.     

Late Diagnosis and Entry to Care after Diagnosis of Human Immunodeficiency Virus Infection: A Country Comparison

Background

Testing for HIV infection and entry to care are the first steps in the continuum of care that benefit individual health and may reduce onward transmission of HIV. We determined the percentage of people with HIV who were diagnosed late and the percentage linked into care overall and by demographic and risk characteristics by country.

Methods

Data were analyzed from national HIV surveillance systems. Six countries, where available, provided data on two late diagnosis indicators (AIDS diagnosis within 3 months of HIV diagnosis, and AIDS diagnosis within 12 months before HIV diagnosis) and linkage to care (≥1 CD4 or viral load test result within 3 months of HIV diagnosis) for people diagnosed with HIV in 2009 or 2010 (most recent year data were available).

Principal Findings

The percentage of people presenting with late stage disease at HIV diagnosis varied by country, overall with a range from 28.7% (United States) to 8.8% (Canada), and by transmission categories. The percentage of people diagnosed with AIDS who had their initial HIV diagnosis within 12 months before AIDS diagnosis varied little among countries, except the percentages were somewhat lower in Spain and the United States. Overall, the majority of people diagnosed with HIV were linked to HIV care within 3 months of diagnosis (more than 70%), but varied by age and transmission category.

Conclusions

Differences in patterns of late presentation at HIV diagnosis among countries may reflect differences in screening practices by providers, public health agencies, and people with HIV. The percentage of people who received assessments of immune status and viral load within 3 months of diagnosis was generally high.     

On the identifiability of metabolic network models

A major problem for the identification of metabolic network models is parameter identifiability, that is, the possibility to unambiguously infer the parameter values from the data. Identifiability problems may be due to the structure of the model, in particular implicit dependencies between the parameters, or to limitations in the quantity and quality of the available data. We address the detection and resolution of identifiability problems for a class of pseudo-linear models of metabolism, so-called linlog models. Linlog models have the advantage that parameter estimation reduces to linear or orthogonal regression, which facilitates the analysis of identifiability. We develop precise definitions of structural and practical identifiability, and clarify the fundamental relations between these concepts. In addition, we use singular value decomposition to detect identifiability problems and reduce the model to an identifiable approximation by a principal component analysis approach. The criterion is adapted to real data, which are frequently scarce, incomplete, and noisy. The test of the criterion on a model with simulated data shows that it is capable of correctly identifying the principal components of the data vector. The application to a state-of-the-art dataset on central carbon metabolism in Escherichia coli yields the surprising result that only $4$ out of $31$ reactions, and $37$ out of $100$ parameters, are identifiable. This underlines the practical importance of identifiability analysis and model reduction in the modeling of large-scale metabolic networks. Although our approach has been developed in the context of linlog models, it carries over to other pseudo-linear models, such as generalized mass-action (power-law) models. Moreover, it provides useful hints for the identifiability analysis of more general classes of nonlinear models of metabolism.     

On testing structural identifiability by a simple scaling method: Relying on scaling symmetries can be misleading

A recent paper published in PLOS Computational Biology [1] introduces the Scaling Invariance Method (SIM) for analysing structural local identifiability and observability. These two properties define mathematically the possibility of determining the values of the parameters (identifiability) and states (observability) of a dynamic model by observing its output. In this note we warn that SIM considers scaling symmetries as the only possible cause of non-identifiability and non-observability. We show that other types of symmetries can cause the same problems without being detected by SIM, and that in those cases the method may lead one to conclude that the model is identifiable and observable when it is actually not.     

Bone tissue,lyophilized and stored at room temperature for 15 days or more,is not capable of transmitting HIV,HCV or HBV

Over the past 57 years, 17 recipients of frozen bone have been infected with: HIV (Centers for Disease Control and Prevention in Morb Mortal Wkly Rep MMWR 37(39):597–599, 1988; Li et al. in J Formos Med Assoc 100(5):350–351, 2001; Simonds et al. in NEJM 326(11):726–732, 1992; Schratt et al. in Unfallchirurg 99(9):679–684, 1996); HCV (Eggen and Nordbo in NEJM 326(6):411, 1992; Conrad et al. in J Bone Joint Surg Am 77:214–224, 1995; Trotter in J Bone Joint Surg Am 851(11):2215–2217, 2003; Tugwell et al. in Ann of Internal Med 143(9):648–654, 2005); or HBV (Shutkin in J Bone Joint Surg Am 36:160–162, 1954). However, bone, lyophilized and stored at room temperature, has never transmitted these viral diseases. A literature review was undertaken to determine whether there is any evidence that lyophilized bone is capable of transmitting HIV, HCV and HBV.     

Effectiveness of commercial inhibitors against subtype F HIV-1 protease

Subtype F wild type HIV protease has been kinetically characterized using six commercial inhibitors (amprenavir, indinavir, lopinavir, nelfinavir, ritonavir and saquinavir) commonly used for HIV/AIDS treatment, as well as inhibitor TL-3 and acetyl-pepstatin. We also obtained kinetic parameters for two multi-resistant proteases (one of subtype B and one of subtype F) harboring primary and secondary mutations selected by intensive treatment with ritonavir/nelfinavir. This newly obtained biochemical data shows that all six studied commercially available protease inhibitors are significantly less effective against subtype F HIV proteases than against HIV proteases of subtype B, as judged by increased K_i and biochemical fitness (vitality) values. Comparison with previously reported kinetic values for subtype A and C HIV proteases show that subtype F wild type proteases are significantly less susceptible to inhibition. These results demonstrate that the accumulation of natural polymorphisms in subtype F proteases yields catalytically more active enzymes with a large degree of cross-resistance, which thus results in strong virus viability.