Declarative Representation of Uncertainty in Mathematical Models

Citation: Miller AK, Britten RD, Nielsen PMF ( Declarative Representation of Uncertainty in Mathematical Models Andrew K. Miller 0 Randall D. Britten 0 Poul M. F. Nielsen 0 Filippo Castiglione, National Research Council of Italy (CNR), Italy 0 1 Auckland Bioengineering Institute, University of Auckland , Auckland , New Zealand , 2 Department of Engineering Science, Faculty of Engineering, University of Auckland , Auckland , New Zealand An important aspect of multi-scale modelling is the ability to represent mathematical models in forms that can be exchanged between modellers and tools. While the development of languages like CellML and SBML have provided standardised declarative exchange formats for mathematical models, independent of the algorithm to be applied to the model, to date these standards have not provided a clear mechanism for describing parameter uncertainty. Parameter uncertainty is an inherent feature of many real systems. This uncertainty can result from a number of situations, such as: when measurements include inherent error; when parameters have unknown values and so are replaced by a probability distribution by the modeller; when a model is of an individual from a population, and parameters have unknown values for the individual, but the distribution for the population is known. We present and demonstrate an approach by which uncertainty can be described declaratively in CellML models, by utilising the extension mechanisms provided in CellML. Parameter uncertainty can be described declaratively in terms of either a univariate continuous probability density function or multiple realisations of one variable or several (typically non-independent) variables. We additionally present an extension to SED-ML (the Simulation Experiment Description Markup Language) to describe sampling sensitivity analysis simulation experiments. We demonstrate the usability of the approach by encoding a sample model in the uncertainty markup language, and by developing a software implementation of the uncertainty specification (including the SED-ML extension for sampling sensitivty analyses) in an existing CellML software library, the CellML API implementation. We used the software implementation to run sampling sensitivity analyses over the model to demonstrate that it is possible to run useful simulations on models with uncertainty encoded in this form. - Funding: This work was funded by the Virtual Physiological Human Share project (http://www.vph-share.eu), under European Framework 7. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing Interests: The authors have declared that no competing interests exist. Declarative model representation languages provide a significant opportunity for improving multi-scale modelling workflows, because they cleanly separate the description of the mathematical problem from any algorithmic description, and do so in a way that allows smaller models to be easily composed to build large multiscale models. Declarative model representation languages are best understood through comparison to imperative languages; imperative languages describe a series of steps taken to perform some computation, while models in declarative languages simply make assertions (as is typically done in descriptions of models in academic literature), leaving the numerical application of those assertions up to software packages. This approach has the important benefit that the same model can be used for multiple purposes. For example, a description of some ordinary differential equations and their initial values (an ODE-IV problem) might be used to render equations for a manuscript, solve the ODE-IV problem numerically to understand the time evolution of the system, be used to compute an analytic Jacobian or analytic solution using another solver package, be used in a sensitivity analysis, and be composed into a large multi-scale model, all without reformulating the model. A number of declarative mathematical model representation languages exist in the literature; many of them have been developed with particular problem domains in mind. For example, Systems Biology Markup Language, or SBML [1] allows mathematical models to be described, with a focus on systems biology. CellML [2,3] is an example of a modelling language which has been designed to be domain neutral. The CellML project hosts a repository of CellML models [4] containing, at the time of writing, 557 workspaces, each of which contains one or more related models (mostly drawn from various fields of biology). CellML is also one of the modelling languages selected for use in the European Framework 7 Virtual Physiological Human project. For these reasons, this paper uses CellML as the starting point for representing uncertainty in mathematical models. However, most of what is presented here could be adapted to other declarative languages. Uncertainty in model parameters can arise from diverse sources. A parameter may have been measured experimentally, yielding information about the value of the parameter, but not an exact value. Often, there may be a statistical model describing prior distributions and the relationship between samples (and the random variables from which they are sampled) and the particular parameterisation used in an experiment; the posterior distribution of the parameters can then be computed either analytically or using numerical methods (such as BUGS, Bayesian Inference Using Gibbs Sampling [5] and subsequent refinements). Another common source of uncertainty is where there is no experimental data available for a parameter, but due to physical and other constraints, a modeller has an idea of the range of values in which a parameter lies. Modellers will often be able to suggest a subjective probability distribution for the parameter; for example, a modeller who knows that a parameter value must fall in the interval (a, b) may postulate, a priori, that the true value is uniformly likely to be any value between a and b. A further common source of uncertainty arises when producing models of individuals from a population. Each individual may have a specific fixed value of a parameter, with variation of the parameter across the population; if a particular parameter has not been measured in a particular individual, the parameter is uncertain in an individual-specific model. ODE-IV problems with uncertain parameters are distinct from stochastic differential equation initial value (SDE-IV) problems. SDE-IV problems contain references to stochastic functions that vary with time, while the class of problem described here describes parameters with a single but unknown true value that holds for all time values. For uncertainty information to be useful in a declarative model, some representation for the posterior distribution of uncertain parameters is required. We will briefly summarise the existing litera (...truncated)