DAISY: A new software tool to test global identifiability of biological and physiological systems
Introduction
Parameters characterizing the behavior of unobservable features of biological and physiological systems, e.g. the effect of a drug on its target organ, are usually not amenable to direct measurement. Their measurement is thus usually approached indirectly as a parameter estimation problem [1], [13]. A dynamic model describing the internal structure of the system is formulated theoretically, based on physical, chemical, conservation and transport laws, e.g. of enzyme kinetics and pharmacokinetics. In general, these physiological models take the mathematical form of linear or nonlinear dynamic state-space models, depending on unknown parameters. Usually these parameters are unknown and cannot be pre-specified, and need to be estimated from data collected experimentally by measuring the observable model variables (inputs and outputs). In order to solve the estimation problem, input–output (I/O) experiments are designed. A fundamental prerequisite for parameter estimation to be well posed is global identifiability of the parametric model (see, e.g. [1], [5], [8], [13]). This property states that, under ideal conditions of noise-free observations and error-free model structure, the unknown parameters of the postulated model can be uniquely (and exactly) recovered from the knowledge of the input–output variables of the designed input–output experiment.
Note that the property of a priori identifiability regards an ideal context of error-free model structure and noise-free measurements and thus it is a necessary, but not a sufficient condition to ensure that an accurate identification of the model parameters from real input–output data is possible (e.g. data may be too noisy or the problem too badly conditioned, etc.). However, if the parameters of the postulated model are not uniquely identifiable, even in the theoretical most favorable situation, they will never be identifiable in a practical experiment where model structure misspecification and noise in the measurements are inevitably present. Without a guarantee of a priori identifiability, the estimates of the parameters which could, nevertheless, sometimes be obtained by some numerical optimization algorithms, will be totally unreliable and random. Unfortunately, despite its essential role in parameter identification, identifiability analysis has often been neglected by many researchers. Use of a non uniquely identifiable model in a clinical setting, may possibly compromise the distinction of the normal versus pathological state, ultimately leading the researcher physician to draw potentially erroneous conclusions.
Identifiability also impacts on the design of experiments, by providing guidelines on the selection of input and output sites to allow unique identifiability [12]. This is particularly useful when dealing with intact physiological systems, where both the number and the location of possible inputs and outputs is often very limited. It has been shown that a priori identifiability results can be used to achieve the formulation of a minimal, i.e. necessary and sufficient, input–output configuration for complex experimental design.
Identifiability analysis can be helpful also to provide guidelines to deal with non-identifiability, either providing hints on how to simplify the model structure or indicating when more information (measured data) are needed for the specific experiment.
Identifiability analysis of nonlinear systems is in general very difficult. The need for such a theory is unquestionable as dealing with nonlinear models, i.e. the Michaelis–Menten equation, is very common in modelling say enzyme kinetics and drug metabolism. Unfortunately its applicability has been seriously hampered by the heavy computational burden of the available techniques. Specifically, the problem translates mathematically into checking solvability of an unusually large system of nonlinear algebraic equations. The number of equations and their degree generally increases with the model order. Note that identifiability of the linearized model can provide information on the identifiability of the original nonlinear model only under certain restrictions [4] but, in general, this is not true and the identifiability of nonlinear models has to be tackled directly.
Different approaches have been proposed in literature [1], [2], [5], [4], [8], [9], [14] but, to the best of our knowledge, no general software tools currently exist to perform the identifiability analysis for a nonlinear dynamic model, despite the crucial importance of this step in the modeling process.
As our contribution to the solution of this problem, we have developed a new differential algebra algorithm [2], [11] which integrates the different strategies proposed in Refs. [8], [9] and broadens their domain of applicability to models described by (linear and) nonlinear models involving polynomial or rational functions, and are initialized at either unknown or known initial conditions. We refer the reader to [2], [11] where this algorithm is described in detail. In this paper we shall present a new software tool, DAISY (Differential Algebra for Identifiability of SYstems) implementing the algorithm in the symbolic language REDUCE version 3.8. The goal of this new software tool, which we propose to make broadly available, is to bring to the field of biomedical applications a piece of software which, although being based on a rather sophisticated set of mathematical tools, will not require knowledge of higher mathematics and computer algebra by the user and yet will allow him to tackle problems which are hard and computationally intensive in a transparent way, without requiring any knowledge of high-level programming languages.
The layout of the paper is the following:
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In Section 2, the basic dynamic structure of the models under test is introduced. Some definitions regarding parameter identifiability are recalled and the identifiability test based on the characteristic set is briefly illustrated.
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In Section 3, a simple example to illustrate the differential algebra method is presented.
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In Section 4, the algorithm and its features are described.
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In Section 5, the software tool DAISY is presented.
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In Section 6, a case of usage of the software DAISY for the analysis of a priori global identifiability of a biomedical model is presented in some detail. In particular, the input file that the user has to provide to DAISY and the corresponding output file with the identifiability results are reported.
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Some basic concepts of differential algebra useful for identifiability analysis test are recalled in Appendix A. In particular, differential ideals and the characteristic set are defined together with their principal properties.
Section snippets
Checking a priori identifiability
This section provides the reader with a brief description of the theory behind our software tool. Such an overview is not meant to be complete. A detailed documentation of the theory is reported in Refs. [2], [11].
Analysis of an example
Here we shall analyze in mathematical notation, a nonlinear model for which the calculations can be done by hand. This is meant to illustrate the basic steps of the differential algebra identifiability algorithm described in the previous section. Another example analyzed via the software tool, will be presented in Section 6.
Example 1 Consider a two-compartment model with Michaelis–Menten kinetics. The model structure, together with its input–output configuration, is shown in Fig. 1. The model is
The DAISY (Differential Algebra for Identifiability of SYstems) algorithm
The input of the algorithm is provided by the differential polynomials defining the dynamic system (1), (2), the number of inputs and outputs, the ranked list of input, output and state variables, the list of the unknown parameters and, if present, the equality constraints among the parameters (3).
The algorithm consists of the following sequence of steps:
- 0.
If one or more polynomials are rational, they are reduced to the same denominator (this can be easily accomplished using the built-in REDUCE
The software tool DAISY
The software tool implementing the above algorithm is written in REDUCE version 3.8. REDUCE 3.8 is an interactive program designed for general algebraic computations of interest to mathematicians, scientists and engineers. The main aim of REDUCE is to support algebra calculations that are not feasible by hand. The REDUCE computer algebra system is not public domain, but it is available at a small fee. The identifiability software tool DAISY is available at the web site //www.dei.unipd.it/~pia/
A case study
DAISY has been tested and evaluated “in house” on a representative set of biological and biomedical cases. To check global identifiability with DAISY, the dynamical model should be provided in a separate file according to a specific scheme. We shall illustrate this scheme by giving an example.
We will use a very common model in enzyme kinetics and drug metabolism, i.e. the two-compartment open model with Michaelis–Menten elimination. The model structure, together with its input–output
Conclusions
In this paper we have described DAISY (Differential Algebra for Identifiability of SYstems), a general software tool allowing biomedical researchers to perform global identifiability analysis for linear and nonlinear dynamic models. In particular, DAISY effectively facilitates the solution to the underappreciated problem of determining if unique parameter estimation from the experimental data is theoretically possible. Although DAISY is a computer-algebra code implementing a differential
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