Structural identifiability of dynamic systems biology models

22 páginas, 5 figuras, 2 tablas.-- This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.A powerful way of gaining insight into biological systems is by creating a nonlinear differential equation model, which usually contains many unknown parameters. Such a model is called structurally identifiable if it is possible to determine the values of its parameters from measurements of the model outputs. Structural identifiability is a prerequisite for parameter estimation, and should be assessed before exploiting a model. However, this analysis is seldom performed due to the high computational cost involved in the necessary symbolic calculations, which quickly becomes prohibitive as the problem size increases. In this paper we show how to analyse the structural identifiability of a very general class of nonlinear models by extending methods originally developed for studying observability. We present results about models whose identifiability had not been previously determined, report unidentifiabilities that had not been found before, and show how to modify those unidentifiable models to make them identifiable. This method helps prevent problems caused by lack of identifiability analysis, which can compromise the success of tasks such as experiment design, parameter estimation, and model-based optimization. The procedure is called STRIKE-GOLDD (STRuctural Identifiability taKen as Extended-Generalized Observability with Lie Derivatives and Decomposition), and it is implemented in a MATLAB toolbox which is available as open source software. The broad applicability of this approach facilitates the analysis of the increasingly complex models used in systems biology and other areasAFV acknowledges funding from the Galician government (Xunta de Galiza, Consellería de Cultura, Educación e Ordenación Universitaria http://www.edu.xunta.es/portal/taxonomy/term/206) through the I2C postdoctoral program, fellowship ED481B2014/133-0. AB and AFV were partially supported by grant DPI2013-47100-C2-2-P from the Spanish Ministry of Economy and Competitiveness (MINECO). AFV acknowledges additional funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 686282 (CanPathPro). AP was partially supported through EPSRC projects EP/M002454/1 and EP/J012041/1.Peer reviewe

Dynamical compensation and structural identifiability: analysis, implications, and reconciliation

The concept of dynamical compensation has been recently introduced to describe the ability of a biological system to keep its output dynamics unchanged in the face of varying parameters. Here we show that, according to its original definition, dynamical compensation is equivalent to lack of structural identifiability. This is relevant if model parameters need to be estimated, which is often the case in biological modelling. This realization prompts us to warn that care should we taken when using an unidentifiable model to extract biological insight: the estimated values of structurally unidentifiable parameters are meaningless, and model predictions about unmeasured state variables can be wrong. Taking this into account, we explore alternative definitions of dynamical compensation that do not necessarily imply structural unidentifiability. Accordingly, we show different ways in which a model can be made identifiable while exhibiting dynamical compensation. Our analyses enable the use of the new concept of dynamical compensation in the context of parameter identification, and reconcile it with the desirable property of structural identifiability

Mathematical deconvolution of CAR T-cell proliferation and exhaustion from real-time killing assay data.

Chimeric antigen receptor (CAR) T-cell therapy has shown promise in the treatment of haematological cancers and is currently being investigated for solid tumours, including high-grade glioma brain tumours. There is a desperate need to quantitatively study the factors that contribute to the efficacy of CAR T-cell therapy in solid tumours. In this work, we use a mathematical model of predator-prey dynamics to explore the kinetics of CAR T-cell killing in glioma: the Chimeric Antigen Receptor T-cell treatment Response in GliOma (CARRGO) model. The model includes rates of cancer cell proliferation, CAR T-cell killing, proliferation, exhaustion, and persistence. We use patient-derived and engineered cancer cell lines with an in vitro real-time cell analyser to parametrize the CARRGO model. We observe that CAR T-cell dose correlates inversely with the killing rate and correlates directly with the net rate of proliferation and exhaustion. This suggests that at a lower dose of CAR T-cells, individual T-cells kill more cancer cells but become more exhausted when compared with higher doses. Furthermore, the exhaustion rate was observed to increase significantly with tumour growth rate and was dependent on level of antigen expression. The CARRGO model highlights nonlinear dynamics involved in CAR T-cell therapy and provides novel insights into the kinetics of CAR T-cell killing. The model suggests that CAR T-cell treatment may be tailored to individual tumour characteristics including tumour growth rate and antigen level to maximize therapeutic benefit

Input-dependent structural identifiability of nonlinear systems

A dynamic model is structurally identifiable if it is possible to infer its unknown parameters by observing its output. Structural identifiability depends on the system dynamics, output, and input, as well as on the specific values of initial conditions and parameters. Here we present a symbolic method that characterizes the input that a model requires to be structurally identifiable. It determines which derivatives must be non-zero in order to have a sufficiently exciting input. Our approach considers structural identifiability as a generalization of nonlinear observability and incorporates extended Lie derivatives. The methodology assesses structural identifiability for time-varying inputs and, additionally, it can be used to determine the input profile that is required to make the parameters structurally locally identifiable. Furthermore, it is sometimes possible to replace an experiment with time-varying input with multiple experiments with constant inputs. We implement the resulting method as a MATLAB toolbox named STRIKE-GOLDD2. This tool can assist in the design of new experiments for the purpose of parameter estimation

Analysis of a Cardiac-Necrosis-Biomarker Release in Patients with Acute Myocardial Infarction via Nonlinear Mixed-Effects Models

The release of the cardiac troponin T (cTnT) in patients with acute myocardial infarc tion (AMI) has been analyzed through a methodology based on nonlinear mixed-effects (NME) models. The aim of this work concerns the investigation of any possible relationship between clin ical covariates and the dynamics of the release of cTnT to derive more detailed and useful clinical information for the correct treatment of these patients. An ad-hoc mechanistic model describing the biomarker release process after AMI has been devised, assessed, and exploited to evaluate the im pact of the available clinical covariates on the cTnT release dynamic. The following approach was tested on a preliminary dataset composed of a small number of potential clinical covariates: em ploying an unsupervised approach, and despite the limited sample size, dyslipidemia, a known risk factor for cardiovascular disease, was found to be a statistically significant covariate. By increasing the number of covariates considered in the model, and patient cohort, we envisage that this approach may provide an effective means to automatically classify AMI patients and to investigate the role of interactions between clinical covariates and cTnT relea

Observability, Identifiability and Epidemiology -- A survey

In this document we introduce the concepts of Observability and Iden-tifiability in Mathematical Epidemiology. We show that, even for simple and well known models, these properties are not always fulfilled. We also consider the problem of practical observability and identi-fiability which are connected to sensitivity and numerical condition numbers

Structural identifiability of compartmental models for infectious disease transmission is influenced by data type

If model identifiability is not confirmed, inferences from infectious disease transmission models may not be reliable, so they might result in misleading recommendations. Structural identifiability analysis characterises whether it is possible to obtain unique solutions for all unknown model parameters, given the model structure. In this work, we studied the structural identifiability of some typical deterministic compartmental models for infectious disease transmission, focusing on the influence of the data type considered as model output on the identifiability of unknown model parameters, including initial conditions. We defined 26 model versions, each having a unique combination of underlying compartmental structure and data type(s) considered as model output(s). Four compartmental model structures and three common data types in disease surveillance (incidence, prevalence and detected vector counts) were studied. The structural identifiability of some parameters varied depending on the type of model output. In general, models with multiple data types as outputs had more structurally identifiable parameters, than did models with a single data type as output. This study highlights the importance of a careful consideration of data types as an integral part of the inference process with compartmental infectious disease transmission models

Structural identifiability of large systems biology models

A fundamental principle of systems biology is its perpetual need for new technologies that can solve challenging biological questions. This precept will continue to drive the development of novel analytical tools. The virtuous cycle of biological progress can therefore only exist when experts from different disciplines including biology, chemistry, computer science, engineering, mathematics, and medicine collaborate. General opinion is however that one of the challenges facing the systems biology community is the lag in the development of such technologies. The topic of structural identifiability in particular has been of interest to the systems biology community. This is because researchers in this field often face experimental limitations. These limitations, combined with the fact that systems biology models can contain vast numbers of unknown parameters, necessitate an identifiability analysis. In reality, analysing the structural identifiability of systems biology models, even when they contain only a few states and system parameters, may be challenging. As these models increase in size and complexity, this difficulty is exasperated, and one becomes limited to only a few methods capable of analysing large ordinary differential equation models. In this thesis I study the use of a computationally efficient algorithm, well suited to the analysis of large models, in the model development process. The three related objectives of this thesis are: 1) develop an accurate method to asses the structural identifiability of large possibly nonlinear ordinary differential models, 2) implement thismethod in the preliminary design of experiments, and 3) use the method to address the topic of structural unidentifiability. To improve the method’s accuracy, I systematically study the role of individual factors, such as the number of experimentally measured sensors, on the sharpness of results. Based on the findings, I propose measures that can improve numerical accuracy. To address the second objective, I introduce an iterative identifiability algorithm that can determine minimal sets of outputs that need to be measured to ensure a model’s local structural identifiability. I also illustrate how one could potentially reduce the computational demand of the algorithm, enabling a user to detect minimal output sets of large ordinary differential equation models within minutes. For the last objective, I investigate the role of initial conditions in a model’s structural unidentifiability. I show that the method can detect problematic values for large ordinary differential equation models. I illustrate its role in reinstating the local structural identifiability of a model by identifying problematic initial conditions. I also show that the method can provide theoretical suggestions for the reparameterization of structurally unidentifiable models. The novelty of this work is that the algorithm allows for unknown initial conditions to be parameterised and accordingly, repameterisations requiring the transformation of states, associated with unidentifiable initial conditions, can easily be obtained. The computational efficiency of the method allows for the reparameterisation of large ordinary differential equation models in particular. To conclude, in this thesis I introduce an method that can be used during the model development process in an array of useful applications. These include: 1) determining minimal output sets, 2) reparameterising structurally unidentifiable models and 3) detecting problematic initial conditions. Each of these application can be implemented before any experiments are conducted and can play a potential role in the optimisation of the modelling process