Review: to be or not to be an identifiable model. Is this a relevant question in animal science modelling?

International audienceWhat is a good (useful) mathematical model in animal science? For models constructed for prediction purposes, the question of model adequacy (usefulness) has been traditionally tackled by statistical analysis applied to observed experimental data relative to model-predicted variables. However, little attention has been paid to analytic tools that exploit the mathematical properties of the model equations. For example, in the context of model calibration, before attempting a numerical estimation of the model parameters, we might want to know if we have any chance of success in estimating a unique best value of the model parameters from available measurements. This question of uniqueness is referred to as structural identifiability; a mathematical property that is defined on the sole basis of the model structure within a hypothetical ideal experiment determined by a setting of model inputs (stimuli) and observable variables (measurements). Structural identifiability analysis applied to dynamic models described by ordinary differential equations (ODE) is a common practice in control engineering and system identification. This analysis demands mathematical technicalities that are beyond the academic background of animal science, which might explain the lack of pervasiveness of identifiability analysis in animal science modelling. To fill this gap, in this paper we address the analysis of structural identifiability from a practitioner perspective by capitalizing on the use of dedicated software tools. Our objectives are (i) to provide a comprehensive explanation of the structural identifiability notion for the community of animal science modelling, (ii) to assess the relevance of identifiability analysis in animal science modelling and (iii) to motivate the community to use identifiability analysis in the modelling practice (when the identifiability question is relevant). We focus our study on ODE models. By using illustrative examples that include published mathematical models describing lactation in cattle, we show how structural identifiability analysis can contribute to advancing mathematical modelling in animal science towards the production of useful models and highly informative experiments. Rather than attempting to impose a systematic identifiability analysis to the modelling community during model developments, we wish to open a window towards the discovery of a powerful tool for model construction and experiment design

Identifying Bayesian Optimal Experiments for Uncertain Biochemical Pathway Models

Pharmacodynamic (PD) models are mathematical models of cellular reaction networks that include drug mechanisms of action. These models are useful for studying predictive therapeutic outcomes of novel drug therapies in silico. However, PD models are known to possess significant uncertainty with respect to constituent parameter data, leading to uncertainty in the model predictions. Furthermore, experimental data to calibrate these models is often limited or unavailable for novel pathways. In this study, we present a Bayesian optimal experimental design approach for improving PD model prediction accuracy. We then apply our method using simulated experimental data to account for uncertainty in hypothetical laboratory measurements. This leads to a probabilistic prediction of drug performance and a quantitative measure of which prospective laboratory experiment will optimally reduce prediction uncertainty in the PD model. The methods proposed here provide a way forward for uncertainty quantification and guided experimental design for models of novel biological pathways

Delineating Parameter Unidentifiabilities in Complex Models

Scientists use mathematical modelling to understand and predict the properties of complex physical systems. In highly parameterised models there often exist relationships between parameters over which model predictions are identical, or nearly so. These are known as structural or practical unidentifiabilities, respectively. They are hard to diagnose and make reliable parameter estimation from data impossible. They furthermore imply the existence of an underlying model simplification. We describe a scalable method for detecting unidentifiabilities, and the functional relations defining them, for generic models. This allows for model simplification, and appreciation of which parameters (or functions thereof) cannot be estimated from data. Our algorithm can identify features such as redundant mechanisms and fast timescale subsystems, as well as the regimes in which such approximations are valid. We base our algorithm on a novel quantification of regional parametric sensitivity: multiscale sloppiness. Traditionally, the link between parametric sensitivity and the conditioning of the parameter estimation problem is made locally, through the Fisher Information Matrix. This is valid in the regime of infinitesimal measurement uncertainty. We demonstrate the duality between multiscale sloppiness and the geometry of confidence regions surrounding parameter estimates made where measurement uncertainty is non-negligible. Further theoretical relationships are provided linking multiscale sloppiness to the Likelihood-ratio test. From this, we show that a local sensitivity analysis (as typically done) is insufficient for determining the reliability of parameter estimation, even with simple (non)linear systems. Our algorithm provides a tractable alternative. We finally apply our methods to a large-scale, benchmark Systems Biology model of NF-$\kappa$B, uncovering previously unknown unidentifiabilities

Mixed Effects Modeling of Deterministic and Stochastic Dynamical Systems - Methods and Applications in Drug Development

Mathematical models based on ordinary differential equations (ODEs) are commonly used for describing the evolution of a system over time. In drug development, pharmacokinetic (PK) and pharmacodynamic (PD) models are used to characterize the exposure and effect of drugs. When developing mathematical models, an important step is to infer model parameters from experimental data. This can be a challenging problem, and the methods used need to be efficient and robust for the modeling to be successful. This thesis presents the development of a set of novel methods for mathematical modeling of dynamical systems and their application to PK-PD modeling in drug development.A method for regularizing the parameter estimation problem for dynamical systems is presented. The method is based on an extension of ODEs to stochastic differential equations (SDEs), which allows for stochasticity in the system dynamics, and is shown to lead to a parameter estimation problem that is easier to solve.The combination of parameter variability and SDEs are investigated, allowing for an additional source of variability compared to the standard nonlinear mixed effects (NLME) model. For NLME models with dynamics described using either ODEs or SDEs, a novel parameter estimation algorithm is presented. The method is a gradient-based optimization method where the exact gradient of the likelihood function is calculated using sensitivity equations, which is shown to give a substantial improvement in computational speed compared to existing methods. The methods developed have been integrated into NLMEModeling, a freely available software package for mixed effects modeling in Wolfram Mathematica. The package allows for general model specifications and offers a user-friendly environment for NLME modeling of dynamical systems.The SDE-NLME framework is used in two applied modeling problems in drug development. First, a previously published PK model of nicotinic acid is extended to incorporate SDEs. By extending the ODE model to an SDE model, it is shown that an additional source of variability can be quantified. Second, the SDE-NLME framework is applied in a model-based analysis of peak expiratory flow (PEF) diary data from two Phase III studies in asthma. The established PEF model can describe several aspects of the PEF dynamics, including long-term fluctuations. The association to exacerbation risk is investigated using a repeated time-to-event model, and several characteristics of the PEF dynamics are shown to be associated with exacerbation risk.The research presented in this doctoral thesis demonstrates the development of a set of methods and applications of mathematical modeling of dynamical systems. In this work, the methods were primarily applied in the field of PK-PD modeling, but are also applicable in other scientific fields

CTP:A Causal Interpretable Model for Non-Communicable Disease Progression Prediction

Non-communicable disease is the leading cause of death, emphasizing the need for accurate prediction of disease progression and informed clinical decision-making. Machine learning (ML) models have shown promise in this domain by capturing non-linear patterns within patient features. However, existing ML-based models cannot provide causal interpretable predictions and estimate treatment effects, limiting their decision-making perspective. In this study, we propose a novel model called causal trajectory prediction (CTP) to tackle the limitation. The CTP model combines trajectory prediction and causal discovery to enable accurate prediction of disease progression trajectories and uncover causal relationships between features. By incorporating a causal graph into the prediction process, CTP ensures that ancestor features are not influenced by the treatment of descendant features, thereby enhancing the interpretability of the model. By estimating the bounds of treatment effects, even in the presence of unmeasured confounders, the CTP provides valuable insights for clinical decision-making. We evaluate the performance of the CTP using simulated and real medical datasets. Experimental results demonstrate that our model achieves satisfactory performance, highlighting its potential to assist clinical decisions. Source code is in \href{https://github.com/DanielSun94/CFPA}{here}.Comment: 25 pages, 5 figures, 12 table

Quantitative Kinetic Models from Intravital Microscopy: A Case Study Using Hepatic Transport

The liver performs critical physiological functions, including metabolizing and removing substances, such as toxins and drugs, from the bloodstream. Hepatotoxicity itself is intimately linked to abnormal hepatic transport, and hepatotoxicity remains the primary reason drugs in development fail and approved drugs are withdrawn from the market. For this reason, we propose to analyze, across liver compartments, the transport kinetics of fluorescein-a fluorescent marker used as a proxy for drug molecules-using intravital microscopy data. To resolve the transport kinetics quantitatively from fluorescence data, we account for the effect that different liver compartments (with different chemical properties) have on fluorescein's emission rate. To do so, we develop ordinary differential equation transport models from the data where the kinetics is related to the observable fluorescence levels by "measurement parameters" that vary across different liver compartments. On account of the steep non-linearities in the kinetics and stochasticity inherent to the model, we infer kinetic and measurement parameters by generalizing the method of parameter cascades. For this application, the method of parameter cascades ensures fast and precise parameter estimates from noisy time traces

A matter of timing : A modelling-based investigation of the dynamic behaviour of reproductive hormones in girls and women

Hypothalamus-hypofyse-gonade aksen er en del av det kvinnelige endokrine systemet, og regulerer evnen til reproduksjon. Hormoner produsert og utskilt fra tre kjertler (hypotalamus, hypofysen, eggstokkene) påvirker hverandre via tilbakemeldingsinteraksjoner, som er nødvendige for å etablere en regelmessig menstruasjonssyklus hos kvinner. Matematiske modeller som forutsier utviklingen av slike hormonkonsentrasjoner og modning av eggstokkfollikler er nyttige verktøy for å forstå menstruasjonssyklusens dynamiske oppførsel. Slike modeller kan for eksempel hjelpe oss med å undersøke patologiske tilstander som endometriose og polycystisk ovariesyndrom. Videre kan de brukes til systematiske undersøkelser av effekten av medikamenter på det kvinnelige endokrine systemet. Derfor kan vi potensielt bruke slike menstruasjonsyklusmodeller som kliniske beslutningsstøttessystemer. Vi trenger modeller som forutsier hormonkonsentrasjoner sammen med modningen av eggstokkfollikler hos enkeltindivider gjennom påfølgende sykluser. Dette for å kunne simulere hormonelle behandlinger som stimulerer vekst av eggstokkfolliklene (eggstokkstimuleringsprotokoller). Her legger jeg fram et forslag til en matematisk menstruasjonsyklusmodell og viser modellens evne til å forutsi resultatet av eggstokkstimuleringsprotokoller. For å kalibrere denne typen modell trenges individuelle tidsseriedata. Innsamling av slike data er tidskrevende, og forutsetter høy grad av engasjement fra deltakerne i studien. Det er derfor viktig å finne brukbare datatyper som er mindre tid- og ressurskrevende å samle inn, og som likevel kan brukes til modellkalibrering. En type data som er enklere å samle inn er tversnittdata. I denne avhandlingen har jeg utviklet en prosedyre for å bruke tversnittpopulasjonsdata i modellens kalibreringsprosess, og viser hvordan en modell kalibrert med tversnittdata kan brukes til å forutsi individuelle resultater ved oppdatering av en del av modellens parametere. I tillegg til det vitenskapelige bidraget, håper jeg at avhandlingen min skaper oppmerksomhet rundt viktigheten av forskning på kvinners reproduktive helse, og at avhandlingen underbygger verdien av matematiske modeller i forskning på kvinnehelse.The hypothalamic-pituitary-gonadal axis (HPG axis), a part of the human endocrine system, regulates the female reproductive function. Feedback interactions between hormones secreted from the glands forming the HPG axis are essential for establishing a regular menstrual cycle. Mathematical models predicting the time evolution of hormone concentrations and the maturation of ovarian follicles are useful tools for understanding the dynamic behaviour of the menstrual cycle. Such models can, for example, help us to investigate pathological conditions, such as endometriosis or Polycystic Ovary Syndrome. Furthermore, they can be used to systematically study the effects of drugs on the endocrine system. In doing so, menstrual cycle models could potentially be integrated into clinical routines as clinical decision support systems. For the simulation-based investigation of hormonal treatments aiming to stimulate the growth of ovarian follicles (Controlled Ovarian Stimulation (COS)), we need models that predict hormone concentrations and the maturation of ovarian follicles in biological units throughout consecutive cycles. Here, I propose such a mechanistic menstrual cycle model. I also demonstrate its capability to predict the outcome of COS. Individual time series data is usually used to calibrate mechanistic models having clinical implications. Collecting these data, however, is time-consuming and requires a high commitment from study participants. Therefore, integrating different data sets into the model calibration process is of interest. One type of data that is often more feasible to collect than individual time series is cross-sectional data. As part of my thesis, I developed a workflow based on Bayesian updating to integrate cross-sectional data into the model calibration process. I demonstrate the workflow using a mechanistic model describing the time evolution of reproductive hormones during puberty in girls. Exemplary, I show that a model calibrated with cross-sectional data can be used to predict individual dynamics after updating a subset of model parameters. In addition to the scientific contributions of this thesis, I hope that it creates attention for the importance of research in the area of women's reproductive health and underpins the value of mathematical modelling for this field.Doktorgradsavhandlin

Methods for Reconstructing Networks with Incomplete Information.

Network representations of complex systems are widespread and reconstructing unknown networks from data has been intensively researched in statistical and scientific communities more broadly. Two challenges in network reconstruction problems include having insufficient data to illuminate the full structure of the network and needing to combine information from different data sources. Addressing these challenges, this thesis contributes methodology for network reconstruction in three respects. First, we consider sequentially choosing interventions to discover structure in directed networks focusing on learning a partial order over the nodes. This focus leads to a new model for intervention data under which nodal variables depend on the lengths of paths separating them from intervention targets rather than on parent sets. Taking a Bayesian approach, we present partial-order based priors and develop a novel Markov-Chain Monte Carlo (MCMC) method for computing posterior expectations over directed acyclic graphs. The utility of the MCMC approach comes from designing new proposals for the Metropolis algorithm that move locally among partial orders while independently sampling graphs from each partial order. The resulting Markov Chains mix rapidly and are ergodic. We also adapt an existing strategy for active structure learning, develop an efficient Monte Carlo procedure for estimating the resulting decision function, and evaluate the proposed methods numerically using simulations and benchmark datasets. We next study penalized likelihood methods using incomplete order information as arising from intervention data. To make the notion of incomplete information precise, we introduce and formally define incomplete partial orders which subsumes the important special case of a known total ordering of the nodes. This special case lies along an information lattice and we study the reconstruction performance of penalized likelihood methods at different points along this lattice. Finally, we present a method for ranking a network's potential edges using time-course data. The novelty is our development of a nonparametric gradient-matching procedure and a related summary statistic for measuring the strength of relationships among components in dynamic systems. Simulation studies demonstrate that given sufficient signal moving using this procedure to move from linear to additive approximations leads to improved rankings of potential edges.PhDStatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113316/1/jbhender_1.pd

Mathematical Models of Androgen Resistance in Prostate Cancer Patients under Intermittent Androgen Suppression Therapy

abstract: Predicting resistant prostate cancer is critical for lowering medical costs and improving the quality of life of advanced prostate cancer patients. I formulate, compare, and analyze two mathematical models that aim to forecast future levels of prostate-specific antigen (PSA). I accomplish these tasks by employing clinical data of locally advanced prostate cancer patients undergoing androgen deprivation therapy (ADT). I demonstrate that the inverse problem of parameter estimation might be too complicated and simply relying on data fitting can give incorrect conclusions, since there is a large error in parameter values estimated and parameters might be unidentifiable. I provide confidence intervals to give estimate forecasts using data assimilation via an ensemble Kalman Filter. Using the ensemble Kalman Filter, I perform dual estimation of parameters and state variables to test the prediction accuracy of the models. Finally, I present a novel model with time delay and a delay-dependent parameter. I provide a geometric stability result to study the behavior of this model and show that the inclusion of time delay may improve the accuracy of predictions. Also, I demonstrate with clinical data that the inclusion of the delay-dependent parameter facilitates the identification and estimation of parameters.Dissertation/ThesisDoctoral Dissertation Applied Mathematics 201

Biological Networks

Networks of coordinated interactions among biological entities govern a myriad of biological functions that span a wide range of both length and time scales—from ecosystems to individual cells and from years to milliseconds. For these networks, the concept “the whole is greater than the sum of its parts” applies as a norm rather than an exception. Meanwhile, continued advances in molecular biology and high-throughput technology have enabled a broad and systematic interrogation of whole-cell networks, allowing the investigation of biological processes and functions at unprecedented breadth and resolution—even down to the single-cell level. The explosion of biological data, especially molecular-level intracellular data, necessitates new paradigms for unraveling the complexity of biological networks and for understanding how biological functions emerge from such networks. These paradigms introduce new challenges related to the analysis of networks in which quantitative approaches such as machine learning and mathematical modeling play an indispensable role. The Special Issue on “Biological Networks” showcases advances in the development and application of in silico network modeling and analysis of biological systems