Model Selection and Mixed-Effects Modeling of HIV Infection Dynamics

Pharmacometrics: The Already-Present Future of Precision Pharmacology

The use of mathematical modeling to represent, analyze, make predictions or providing information on data obtained in drug research and development has made pharmacometrics an area of great prominence and importance. The main purpose of pharmacometrics is to provide information relevant to the search for efficacy and safety improvements in pharmacotherapy. Regulatory agencies have adopted pharmacometrics analysis to justify their regulatory decisions, making those decisions more efficient. Demand for specialists trained in the field is therefore growing. In this review, we describe the meaning, history, and development of pharmacometrics, analyzing the challenges faced in the training of professionals. Examples of applications in current use, perspectives for the future, and the importance of pharmacometrics for the development and growth of precision pharmacology are also presented.

A modified SEIR model with a jump in the transmission parameter applied to COVID-19 data on Wuhan

In December 2019, Wuhan, the capital of Hubei Province, was struck by an outbreak of COVID-19. Numerous studies have been conducted to fit COVID-19 data and make statistical inferences. In applications, functions of the parameters in the model are usually used to assess severity of the outbreak. Because of the strategies applied during the struggle against the pandemic, the trend of the parameters changes abruptly. However, time-varying parameters with a jump have received scant attention in the literature. In this study, a modified SEIR model is proposed to fit the actual situation of the COVID-19 epidemic. In the proposed model, the dynamic propagation system is modified because of the high infectivity during incubation, and a time-varying parametric strategy is suggested to account for the utility of the intervention. A corresponding model selection algorithm based on the information criterion is also suggested to detect the jump in the transmission parameter. A real data analysis based on the COVID-19 epidemic in Wuhan and a simulation study demonstrate the plausibility and validity of the proposed method.

Mindfulness instruction for medication adherence among adolescents and young adults living with HIV: a randomized controlled trial

Adolescents and young adults (AYA) 13-24 years old make up a disproportionate 21% of new HIV diagnoses. Unfortunately, they are less likely to treat HIV effectively, with only 30% achieving viral suppression, limiting efforts to interrupt HIV transmission. Previous work with mindfulness-based stress reduction (MBSR) has shown promise for improving treatment in AYA living with HIV (AYALH). This randomized controlled trial compared MBSR with general health education (HT). Seventy-four 13-24-year-old AYALH conducted baseline data collection and were randomized to nine sessions of MBSR or HT. Data were collected at baseline, post-program (3 months), 6 and 12 months on mindfulness and HIV management [medication adherence (MA), HIV viral load (HIV VL), and CD4]. Longitudinal analyses were conducted. The MBSR arm reported higher mindfulness at baseline. Participants were average 20.5 years old, 92% non-Hispanic Black, 51% male, 46% female, and 3% transgender. Post-program, MBSR participants had greater increases than HT in MA (p = 0.001) and decreased HIV VL (p = 0.052). MBSR participants showed decreased mindfulness at follow-up. Given the significant challenges related to HIV treatment in AYALH, these findings suggest that MBSR may play a role in improving HIV MA and decreasing HIV VL. Additional research is merited to investigate MBSR further for this important population.

Optimized phylogenetic clustering of HIV-1 sequence data for public health applications

Clusters of genetically similar infections suggest rapid transmission and may indicate priorities for public health action or reveal underlying epidemiological processes. However, clusters often require user-defined thresholds and are sensitive to non-epidemiological factors, such as non-random sampling. Consequently the ideal threshold for public health applications varies substantially across settings. Here, we show a method which selects optimal thresholds for phylogenetic (subset tree) clustering based on population. We evaluated this method on HIV-1 pol datasets (n = 14, 221 sequences) from four sites in USA (Tennessee, Washington), Canada (Northern Alberta) and China (Beijing). Clusters were defined by tips descending from an ancestral node (with a minimum bootstrap support of 95%) through a series of branches, each with a length below a given threshold. Next, we used pplacer to graft new cases to the fixed tree by maximum likelihood. We evaluated the effect of varying branch-length thresholds on cluster growth as a count outcome by fitting two Poisson regression models: a null model that predicts growth from cluster size, and an alternative model that includes mean collection date as an additional covariate. The alternative model was favoured by AIC across most thresholds, with optimal (greatest difference in AIC) thresholds ranging 0.007-0.013 across sites. The range of optimal thresholds was more variable when re-sampling 80% of the data by location (IQR 0.008 - 0.016, n = 100 replicates). Our results use prospective phylogenetic cluster growth and suggest that there is more variation in effective thresholds for public health than those typically used in clustering studies.

Parameter identifiability and model selection for sigmoid population growth models

Sigmoid growth models, such as the logistic, Gompertz and Richards' models, are widely used to study population dynamics ranging from microscopic populations of cancer cells, to continental-scale human populations. Fundamental questions about model selection and parameter estimation are critical if these models are to be used to make practical inferences. However, the question of parameter identifiability - whether a data set contains sufficient information to give unique or sufficiently precise parameter estimates - is often overlooked. We use a profile-likelihood approach to explore practical parameter identifiability using data describing the re-growth of hard coral. With this approach, we explore the relationship between parameter identifiability and model misspecification, finding that the logistic growth model does not suffer identifiability issues for the type of data we consider whereas the Gompertz and Richards' models encounter practical non-identifiability issues. This analysis of parameter identifiability and model selection is important because different growth models are in biological modelling without necessarily considering whether parameters are identifiable. Standard practices that do not consider parameter identifiability can lead to unreliable or imprecise parameter estimates and potentially misleading mechanistic interpretations. For example, using the Gompertz model, the estimate of the time scale of coral re-growth is 625 days when we estimate the initial density from the data, whereas it is 1429 days using a more standard approach where variability in the initial density is ignored. While tools developed here focus on three standard sigmoid growth models only, our theoretical developments are applicable to any sigmoid growth model and any appropriate data set. MATLAB implementations of all software are available on GitHub.

WEAK SINDY FOR PARTIAL DIFFERENTIAL EQUATIONS

Sparse Identification of Nonlinear Dynamics (SINDy) is a method of system discovery that has been shown to successfully recover governing dynamical systems from data [6, 39]. Recently, several groups have independently discovered that the weak formulation provides orders of magnitude better robustness to noise. Here we extend our Weak SINDy (WSINDy) framework introduced in [28] to the setting of partial differential equations (PDEs). The elimination of pointwise derivative approximations via the weak form enables effective machine-precision recovery of model coefficients from noise-free data (i.e. below the tolerance of the simulation scheme) as well as robust identification of PDEs in the large noise regime (with signal-to-noise ratio approaching one in many well-known cases). This is accomplished by discretizing a convolutional weak form of the PDE and exploiting separability of test functions for efficient model identification using the Fast Fourier Transform. The resulting WSINDy algorithm for PDEs has a worst-case computational complexity of O ( N D + 1 log ( N ) ) for datasets with N points in each of D + 1 dimensions. Furthermore, our Fourier-based implementation reveals a connection between robustness to noise and the spectra of test functions, which we utilize in an a priori selection algorithm for test functions. Finally, we introduce a learning algorithm for the threshold in sequential-thresholding least-squares (STLS) that enables model identification from large libraries, and we utilize scale invariance at the continuum level to identify PDEs from poorly-scaled datasets. We demonstrate WSINDy's robustness, speed and accuracy on several challenging PDEs. Code is publicly available on GitHub at https://github.com/MathBioCU/WSINDy_PDE.

WEAK SINDy: GALERKIN-BASED DATA-DRIVEN MODEL SELECTION

We present a novel weak formulation and discretization for discovering governing equations from noisy measurement data. This method of learning differential equations from data fits into a new class of algorithms that replace pointwise derivative approximations with linear transformations and variance reduction techniques. Compared to the standard SINDy algorithm presented in [S. L. Brunton, J. L. Proctor, and J. N. Kutz, Proc. Natl. Acad. Sci. USA, 113 (2016), pp. 3932-3937], our so-called weak SINDy (WSINDy) algorithm allows for reliable model identification from data with large noise (often with ratios greater than 0.1) and reduces the error in the recovered coefficients to enable accurate prediction. Moreover, the coefficient error scales linearly with the noise level, leading to high-accuracy recovery in the low-noise regime. Altogether, WSINDy combines the simplicity and efficiency of the SINDy algorithm with the natural noise reduction of integration, as demonstrated in [H. Schaeffer and S. G. McCalla, Phys. Rev. E, 96 (2017), 023302], to arrive at a robust and accurate method of sparse recovery.

Learning differential equation models from stochastic agent-based model simulations

Agent-based models provide a flexible framework that is frequently used for modelling many biological systems, including cell migration, molecular dynamics, ecology and epidemiology. Analysis of the model dynamics can be challenging due to their inherent stochasticity and heavy computational requirements. Common approaches to the analysis of agent-based models include extensive Monte Carlo simulation of the model or the derivation of coarse-grained differential equation models to predict the expected or averaged output from the agent-based model. Both of these approaches have limitations, however, as extensive computation of complex agent-based models may be infeasible, and coarse-grained differential equation models can fail to accurately describe model dynamics in certain parameter regimes. We propose that methods from the equation learning field provide a promising, novel and unifying approach for agent-based model analysis. Equation learning is a recent field of research from data science that aims to infer differential equation models directly from data. We use this tutorial to review how methods from equation learning can be used to learn differential equation models from agent-based model simulations. We demonstrate that this framework is easy to use, requires few model simulations, and accurately predicts model dynamics in parameter regions where coarse-grained differential equation models fail to do so. We highlight these advantages through several case studies involving two agent-based models that are broadly applicable to biological phenomena: a birth-death-migration model commonly used to explore cell biology experiments and a susceptible-infected-recovered model of infectious disease spread.

Learning Equations from Biological Data with Limited Time Samples

Equation learning methods present a promising tool to aid scientists in the modeling process for biological data. Previous equation learning studies have demonstrated that these methods can infer models from rich datasets; however, the performance of these methods in the presence of common challenges from biological data has not been thoroughly explored. We present an equation learning methodology comprised of data denoising, equation learning, model selection and post-processing steps that infers a dynamical systems model from noisy spatiotemporal data. The performance of this methodology is thoroughly investigated in the face of several common challenges presented by biological data, namely, sparse data sampling, large noise levels, and heterogeneity between datasets. We find that this methodology can accurately infer the correct underlying equation and predict unobserved system dynamics from a small number of time samples when the data are sampled over a time interval exhibiting both linear and nonlinear dynamics. Our findings suggest that equation learning methods can be used for model discovery and selection in many areas of biology when an informative dataset is used. We focus on glioblastoma multiforme modeling as a case study in this work to highlight how these results are informative for data-driven modeling-based tumor invasion predictions.

Modeling cell population dynamics

 Quantitative modeling is quickly becoming an integral part of biology, due to the ability of mathematical models and computer simulations to generate insights and predict the behavior of living systems. Single-cell models can be incapable or misleading for inferring population dynamics, as they do not consider the interactions between cells via metabolites or physical contact, nor do they consider competition for limited resources such as nutrients or space. Here we examine methods that are commonly used to model and simulate cell populations. First, we cover simple models where analytic solutions are available, and then move on to more complex scenarios where computational methods are required. Overall, we present a summary of mathematical models used to describe cell population dynamics, which may aid future model development and highlights the importance of population modeling in biology.

Modeling cell population dynamics

 Quantitative modeling is quickly becoming an integral part of biology, due to the ability of mathematical models and computer simulations to generate insights and predict the behavior of living systems. Single-cell models can be incapable or misleading for inferring population dynamics, as they do not consider the interactions between cells via metabolites or physical contact, nor do they consider competition for limited resources such as nutrients or space. Here we examine methods that are commonly used to model and simulate cell populations. First, we cover simple models where analytic solutions are available, and then move on to more complex scenarios where computational methods are required. Overall, we present a summary of mathematical models used to describe cell population dynamics, which may aid future model development and highlights the importance of population modeling in biology.

Short-Term Antiretroviral Treatment Recommendations Based on Sensitivity Analysis of a Mathematical Model for HIV Infection of CD₄⁺Τ Cells

HIV infection is one of the most difficult infections to control and manage. The most recent recommendations to control this infection vary according to the guidelines used (US, European, WHO) and are not patient-specific. Unfortunately, no two individuals respond to infection and treatment quite the same way. The purpose of this paper is to make use of the uncertainty and sensitivity analysis to investigate possible short-term treatment options that are patient-specific. We are able to identify the most significant parameters that are responsible for ART outcome and to formulate some insights into the ART success.

Short- and Long-Term Optimal Control of a Mathematical Model for HIV Infection of CD4+T Cells

The main goal of this study was to develop a theoretical short- and long-term optimal control treatment of HIV infection of [Formula: see text] cells. The aim of the mathematical model used herein is to make the free HIV virus particles in the blood decrease, while administering a treatment that is less toxic to patients. Pontryagin's classical control theory is applied to a mathematical model of HIV infection of [Formula: see text] cells characterized by a system of nonlinear differential equations with the following unknown functions: the concentration of susceptible [Formula: see text] cells, [Formula: see text] cells infected by the HIV viruses and free HIV virus particles in the blood.

Complexity in mathematical models of public health policies: a guide for consumers of models

Sanjay Basu and colleagues explain how models are increasingly used to inform public health policy yet readers may struggle to evaluate the quality of models. All models require simplifying assumptions, and there are tradeoffs between creating models that are more "realistic" versus those that are grounded in more solid data. Indeed, complex models are not necessarily more accurate or reliable simply because they can more easily fit real-world data than simpler models can. Please see later in the article for the Editors' Summary.

Dynamical models of biomarkers and clinical progression for personalized medicine: the HIV context

Mechanistic models, based on ordinary differential equation systems, can exhibit very good predictive abilities that will be useful to build treatment monitoring strategies. In this review, we present the potential and the limitations of such models for guiding treatment (monitoring and optimizing) in HIV-infected patients. In the context of antiretroviral therapy, several biological processes should be considered in addition to the interaction between viruses and the host immune system: the mechanisms of action of the drugs, their pharmacokinetics and pharmacodynamics, as well as the viral and host characteristics. Another important aspect to take into account is clinical progression, although its implementation in such modelling approaches is not easy. Finally, the control theory and the use of intrinsic properties of mechanistic models make them very relevant for dynamic treatment adaptation. Their implementation would nevertheless require their evaluation through clinical trials.

An approach for identifiability of population pharmacokinetic-pharmacodynamic models

Mathematical models are routinely used in clinical pharmacology to study the pharmacokinetic and pharmacodynamic properties of a drug in the body. Identifiability of these models is an important requirement for the success of these clinical studies. Identifiability is classified into two types, structural identifiability related to the structure of the mathematical model and deterministic identifiability which is related to the study design. There are existing approaches for assessment of structural identifiability of fixed-effects models, although their use appears uncommon in the literature. In this study, we develop an informal unified approach for simultaneous assessment of structural and deterministic identifiability for fixed and mixed-effects pharmacokinetic or pharmacokinetic-pharmacodynamic models. This approach uses an information theoretic framework. The method is applied both to simple examples to explore known identifiability properties and to a more complex example to illustrate its utility.CPT: Pharmacometrics & Systems Pharmacology (2013) 2, e49; doi:10.1038/psp.2013.25; advance online publication 19 June 2013.

Modelling HIV-1 2-LTR dynamics following raltegravir intensification

A model of reservoir activation and viral replication is introduced accounting for the production of 2-LTR HIV-1 DNA circles following antiviral intensification with the HIV integrase inhibitor raltegravir, considering contributions of de novo infection events and exogenous sources of infected cells, including quiescent infected cell activation. The model shows that a monotonic increase in measured 2-LTR concentration post intensification is consistent with limited de novo infection primarily maintained by sources of infected cells unaffected by raltegravir, such as quiescent cell activation, while a transient increase in measured 2-LTR concentration is consistent with significant levels of efficient (R0 > 1) de novo infection. The model is validated against patient data from the INTEGRAL study and is shown to have a statistically significant fit relative to the null hypothesis of random measurement variation about a mean. We obtain estimates and confidence intervals for the model parameters, including 2-LTR half-life. Seven of the 13 patients with detectable 2-LTR concentrations from the INTEGRAL study have measured 2-LTR dynamics consistent with significant levels of efficient replication of the virus prior to treatment intensification.

HIV model parameter estimates from interruption trial data including drug efficacy and reservoir dynamics

Mathematical models based on ordinary differential equations (ODE) have had significant impact on understanding HIV disease dynamics and optimizing patient treatment. A model that characterizes the essential disease dynamics can be used for prediction only if the model parameters are identifiable from clinical data. Most previous parameter identification studies for HIV have used sparsely sampled data from the decay phase following the introduction of therapy. In this paper, model parameters are identified from frequently sampled viral-load data taken from ten patients enrolled in the previously published AutoVac HAART interruption study, providing between 69 and 114 viral load measurements from 3-5 phases of viral decay and rebound for each patient. This dataset is considerably larger than those used in previously published parameter estimation studies. Furthermore, the measurements come from two separate experimental conditions, which allows for the direct estimation of drug efficacy and reservoir contribution rates, two parameters that cannot be identified from decay-phase data alone. A Markov-Chain Monte-Carlo method is used to estimate the model parameter values, with initial estimates obtained using nonlinear least-squares methods. The posterior distributions of the parameter estimates are reported and compared for all patients.

A structured and dynamic framework to advance traits-based theory and prediction in ecology

Predicting changes in community composition and ecosystem function in a rapidly changing world is a major research challenge in ecology. Traits-based approaches have elicited much recent interest, yet individual studies are not advancing a more general, predictive ecology. Significant progress will be facilitated by adopting a coherent theoretical framework comprised of three elements: an underlying trait distribution, a performance filter defining the fitness of traits in different environments, and a dynamic projection of the performance filter along some environmental gradient. This framework allows changes in the trait distribution and associated modifications to community composition or ecosystem function to be predicted across time or space. The structure and dynamics of the performance filter specify two key criteria by which we judge appropriate quantitative methods for testing traits-based hypotheses. Bayesian multilevel models, dynamical systems models and hybrid approaches meet both these criteria and have the potential to meaningfully advance traits-based ecology.

ABC-SysBio--approximate Bayesian computation in Python with GPU support

MOTIVATION

The growing field of systems biology has driven demand for flexible tools to model and simulate biological systems. Two established problems in the modeling of biological processes are model selection and the estimation of associated parameters. A number of statistical approaches, both frequentist and Bayesian, have been proposed to answer these questions.

RESULTS

Here we present a Python package, ABC-SysBio, that implements parameter inference and model selection for dynamical systems in an approximate Bayesian computation (ABC) framework. ABC-SysBio combines three algorithms: ABC rejection sampler, ABC SMC for parameter inference and ABC SMC for model selection. It is designed to work with models written in Systems Biology Markup Language (SBML). Deterministic and stochastic models can be analyzed in ABC-SysBio.

AVAILABILITY

http://abc-sysbio.sourceforge.net

Parameter estimation and model selection in computational biology

A central challenge in computational modeling of biological systems is the determination of the model parameters. Typically, only a fraction of the parameters (such as kinetic rate constants) are experimentally measured, while the rest are often fitted. The fitting process is usually based on experimental time course measurements of observables, which are used to assign parameter values that minimize some measure of the error between these measurements and the corresponding model prediction. The measurements, which can come from immunoblotting assays, fluorescent markers, etc., tend to be very noisy and taken at a limited number of time points. In this work we present a new approach to the problem of parameter selection of biological models. We show how one can use a dynamic recursive estimator, known as extended Kalman filter, to arrive at estimates of the model parameters. The proposed method follows. First, we use a variation of the Kalman filter that is particularly well suited to biological applications to obtain a first guess for the unknown parameters. Secondly, we employ an a posteriori identifiability test to check the reliability of the estimates. Finally, we solve an optimization problem to refine the first guess in case it should not be accurate enough. The final estimates are guaranteed to be statistically consistent with the measurements. Furthermore, we show how the same tools can be used to discriminate among alternate models of the same biological process. We demonstrate these ideas by applying our methods to two examples, namely a model of the heat shock response in E. coli, and a model of a synthetic gene regulation system. The methods presented are quite general and may be applied to a wide class of biological systems where noisy measurements are used for parameter estimation or model selection.

Parameter estimation is a key issue in systems biology, as it represents the crucial step to obtaining predictions from computational models of biological systems. This issue is usually addressed by “fitting” the model simulations to the observed experimental data. Such approach does not take the measurement noise into full consideration. We introduce a new method built on the combination of Kalman filtering, statistical tests, and optimization techniques. The filter is well-known in control and estimation theory and has found application in a wide range of fields, such as inertial guidance systems, weather forecasting, and economics. We show how the statistics of the measurement noise can be optimally exploited and directly incorporated into the design of the estimation algorithm in order to achieve more accurate results, and to validate/invalidate the computed estimates. We also show that a significant advantage of our estimator is that it offers a powerful tool for model selection, allowing rejection or acceptance of competing models based on the available noisy measurements. These results are of immediate practical application in computational biology, and while we demonstrate their use for two specific examples, they can in fact be used to study a wide class of biological systems.

Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems

Approximate Bayesian computation (ABC) methods can be used to evaluate posterior distributions without having to calculate likelihoods. In this paper, we discuss and apply an ABC method based on sequential Monte Carlo (SMC) to estimate parameters of dynamical models. We show that ABC SMC provides information about the inferability of parameters and model sensitivity to changes in parameters, and tends to perform better than other ABC approaches. The algorithm is applied to several well-known biological systems, for which parameters and their credible intervals are inferred. Moreover, we develop ABC SMC as a tool for model selection; given a range of different mathematical descriptions, ABC SMC is able to choose the best model using the standard Bayesian model selection apparatus.

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.