**2014 Workshop: Titles and Abstracts for Talks and Practicals
**

**Andrea Arnold** (Case Western/NC State)

**Parameter Estimation in Biochemical Models via EnKF
**

**Abstract:** This tutorial will discuss how the ensemble Kalman filter (EnKF) can be used for blind state and parameter estimation in deterministic dynamical systems. We will focus specifically on metabolic models, in which the numerous model parameters and the concentrations of the metabolites in tissue are to be estimated from concentration data in the blood. Here we adapt the EnKF algorithm, a popular method for addressing similar questions in stochastic and turbulent dynamics, for use with deterministic systems where the numerical approximation error is interpreted as the stochastic drift with variance based on classical error estimates of numerical integrators. The viability of the approach is shown by computed examples, including a metabolic system modeling an ischemic episode in a skeletal muscle.

**H.T. Banks** (NC State)

**Mathematical and Statistical Aspects of Inverse Problems**

**Abstract:** We outline fundamental mathematical and statistical methodologies for inverse or parameter estimation problems. Topics to be discussed include least-squares formulations (ordinary, weighted and generalized least squares), sensitivity equations, asymptotic theory for confidence intervals and uncertainty, use of residual plots to analyze correctness of statistical model formulation. Material will be chosen from Chapter 3 of our recent book: H.T. Banks, S. Hu and W.C. Thompson, Modeling and Inverse Problems in the Presence of Uncertainty, Chapman Hall/CRC Press, Boca Raton, FL, 2014.

**Joe DiStefano III** (UCLA)

**Alternative Dynamic Systems Biology Models and Their Analyses**

**Abstract:** Dynamic systems biology modeling can be viewed as a three-step process, first involving the translation of mechanistic biological information (structural data) into one or more (alternative) graphs and parameterized mathematical relationships (structural models). This is typically followed by mathematical and computational analyses and, eventually, quantification from numerical input-output (I-O) data (parameter estimation). We emphasize here that structural models include specification of experimental input and output ports – the knowledge-limiting features of these models. Structural and numerical (practical) identifiability analysis methods then focus – typically one at a time – on particular structural models (with I-O ports specified), for the purpose of maximizing information-extraction about the biology from those experiments, exposing alternative experiment designs when different I-O ports (different models) are explored. Other methodologies intimately related to identifiability are also available for model analysis, including Fisher information matrix-related approaches.

We discuss this bigger picture in math modeling, with some historical perspective, examples, possibly including a new software tool demo (DISTING). Model selection, simplification and model discrimination/distinguishability and the focus.

* These talk topics are major components of my new textbook “Dynamic Systems Biology Modeling and Simulation”, Academic Press/Elsevier, 2013, 2014 (Kindle and other e- editions also available).

**Marisa Eisenberg** (University of Michigan)

**Introduction to structural and practical identifiability**

**Abstract:** In this talk we will introduce some basic identifiability concepts, including structural and practical identifiability. We will introduce several numerical and analytical methods for evaluating model identifiability, including differential algebra approaches, Fisher information-based approaches, likelihood profiling, among others. We will discuss some of the pros and cons of these different methods and will examine them in the context of example models with real-world applications to medicine and epidemiology.

**Pierre Gremaud** (NC State)

**Uncertainties in blood flow calculations and data**

**Abstract:** In several clinical applications, physicians need access to quantities that are hard and/or expensive to measure. Assume for instance we want to evaluate perfusion in a specific brain territory for a patient for whom only very partial (but easily available) data is at hand.

If data is available for a certain population, one can (i) classify that population into groups of like patients, (ii) identify the above patient as a member of one such group and (iii) apply a local model in that group to get a prediction for the patient.

I will give an elementary introduction to classification and local regression methods that make possible the above approach. These machine learning methods are widely used and yet are poorly understood and barely analyzed. I will describe the strengths and weaknesses of such methods and point to some open problems. Concepts will be illustrated in the context of patient specific cerebral blood flow estimates.

**Alun Lloyd** (NC State)

**Practical: Sensitivity Analysis and Least Squares Parameter Estimation for an Epidemic Model**

**
Abstract:** In this practical we shall derive and implement the sensitivity equations for a simple epidemic model, the SIR model. We shall fit this model to some data from an influenza outbreak at a boarding school and use asymptotic statistical theory to obtain standard errors for the estimated parameters.

**Adam Mahdi** (University of Oxford)

**Introduction to Kalman filtering and parameter estimation
**

**Abstract:** The Kalman filter is one of the most popular data fusion algorithms. It allows an estimation of the unknown, true value of a set of variables from noisy measurements. At present the Kalman filter (and its numerous variants) are applied to virtually every navigation device (e.g. GPS), image processing, economics, biomedical engineering, and many others. The aim of this tutorial is to provide an elementary and intuitive introduction to linear and non-linear Kalman filtering and parameter estimation methods in the context of the Baysian formalism. We will also discuss several important variants of the theory including extended and ensemble Kalman filters. Finally, the application of the methods will be illustrated by simple numerical examples.

**Nikki Meshkat** (NC State)

**Identifiability of Linear Compartmental Models**

**Abstract:** Identifiability concerns finding which unknown parameters of a model can be determined from given input-output data. In this talk, we will focus on structural identifiability, that is, whether the parameters of a model could be identified if perfect input-output data (noise-free and of any duration required) were available. In the context of structural identifiability, if the parameters of a model have a unique or finite number of values given input-output data, then the model and its parameters are said to be identifiable. However, if some subset of the parameters can take on an infinite number of values and yet yield the same input-output data, then the model and this subset of parameters are called unidentifiable. Many linear ODE models, used primarily in Systems Biology, are unidentifiable. We study a particular class of unidentifiable models and find conditions to obtain identifiable reparametrizations of these models. We also examine conditions to obtain local identifiability for this class of models, and in particular, show how identifiability can by determined by simply looking at the graphical structure of these linear compartment models.

**Johnny Ottesen** (Roskilde University)

**Patient specific modeling of ultradian and circadian oscillations in the endocrine HPA-axis and its relation to depression**

**Abstract:** Approximately 10% of people in the Western world experience severe depression during their lifetime and many more a milder form of depression. Recently we have shown that in a dataset consisting of 29 subjects there is a strong correlation between groups of hypocortisolemic depressed, normocortisolemic and hypercortisolemic depressed (diagnosed by classical psychiatric methods) and three corresponding groups which are characterized by a new quantitative index based on frequent measurements of concentration of adrenocorticotropic hormone (ACTH) and cortisol during 24 hours [1]. This supports the common believe that depression is related to malfunctions in the biological endocrine system constituted by the hypothalamus-pituitary-adrenal (HPA) axis [2, 3]. We review various types of models of the HPA-axis along with their pro and con both mathematically and biologically. As a result we pose a novel mathematical model of the HPA-axis capable of showing both circadian as well as ultradian oscillations in ACTH and cortisol hormone concentrations [4]. The patterns generated by the model imitate those observed in the corresponding three groups of data and may even be used for patient specific modelling. To make the model patient-specific a non-linear mixed effect approach will be presented and used [4]. Efficiency and reliability of the parameter estimation method is crucial for the applicability of patient-specific modelling and for the specific case it will be discussed. How patient-specific modelling can provide more refined diagnoses offering a tool for developing individual treatment plans will be discussed.

**References:** [1] Ottesen J.T. Etiology and diagnosis of major depression – a novel quantitative approach. Open Journal of Endocrine and Metabolic Diseases, Vol.3, No.2, pp.8, May, 2013.

[2] Henley D.E., Leendertz J.A., Russell G.M., Wood S.A., Taheri S., Woltersdorf W.W., Lightman S.L. Development of an automated blood sampling sys-tem for use in Humans. J. Med. Eng. Technol. 33(3), 2009, 199–208.

[3] Deuschle M., Schweiger U., Weber B., Gothardt U., Körner A., Schmider J., Standhardt H., Lammers C.-H., Heuser I. Diurnal Activity and Pulsatility of the Hypothalamus-Pituitary-Adrenal System in Male Depressed Patients and Healthy Controls. J. Clin. Endocr. Metab. 82(1), 1997, 234-238.

[4] Gudmand-Hoeyer J., Timmermann S. and Ottesen J.T. Patient-specific modeling of the endocrine HPA-axis and its relation to depression: Ultradian and circadian oscillations. To appear in Mathematical Bioscinces 2014.

**Ralph Smith** (NC State)

**Introduction to Uncertainty Quantification**

**Abstract:** The quantification of uncertainties inherent to models, parameters, and experiments is critical to access the accuracy of predictions. In this presentation, we will discuss basic issues that must be addressed when quantifying uncertainties inherent to physical and biological models. This will be motivated by examples drawn from nuclear power plant design, climate models, and HIV models. Uncertainty quantification is naturally addressed in a Bayesian framework and we will use examples to motivate recent Metropolis and propagation algorithms, which are appropriate for large scale simulation models. We will detail the manner in which previously discussed sensitivity analysis and parameter selection techniques constitute critical steps for uncertainty quantification and we will illustrate the construction of surrogate models for complex simulation codes. Open questions and future research directions will be noted throughout the presentation.