In this paper we present an application of the Buchberger algorithm to the solution of a crucial step in the identification of compartmental models of biological systems from input-output experiments: the a priori identifiability problem.
Linear compartmental models are a class of dynamic models based on mass conservation principles which are widely used in biology and medicine to study the kinetics of substances [4,5]. Compartmental models are mathematically described by ordinary differential equations, where the state variables usually are the compartment masses and the parameters are the transfer rate constants among them. The input-output experiment usually consists in injecting some material into one (or more) compartment and measuring a linear function of some of the state variables.
The a priori structural identifiability of linear compartmental models consists in testing if, under the ideal conditions of noise-free observations and error-free model structure (a priori) and independently of the particular values of the parameters (structurally), the unknown parameters of the model can or can not be uniquelly (globally) estimated from the designed experiment [1-4]. A priori identifiability is clearly a necessary prerequisite for well posedness of parameter estimation. It is also crucial in qualitative experiment design [10], that is selecting the input-output configuration necessary to ensure unique identifiability of the unknown parameters.Various methods to test a priori structural identifiability have been proposed, including the transfer function method [1,2], the transfer function topological method [6], the modal matrix method [2,7], and the similarity transformation method [2,3]. However, whatever is the method used there is the need to solve a system of nonlinear algebraic equations (the identifiability equations) which fast increases in number of unknowns, terms and nonlinearity degree as the model connectivity and/or the number of compartments increase. While some identifiability results are available on specific classes of models, there is no algorithm to test global identifiability of compartmental models of general structure. Recently, a differential algebra algorithm, implemented in Maple, has been proposed [8] for low dimension nonlinear models: however its applicability to linear compartmental models is severely limited by computability bounds (up to three compartments).
The aim of our work has been to develop a new algorithm for checking a priori global identifiability of linear compartmental models of general structure by combining one classical method with computer algebra methods, in particular with the Gröbner basis. In principle all the methods discussed above could be used to generate the identifiability equations, but we have chosen the one which makes the algorithm to calculate the Gröbner basis successful for the largest class of models, i.e. the one which reduces the complexity of the identifiability equations in terms of number of unknowns, number of terms, and nonlinearity degree. The algorithm is essentially a two stage one. First, the transfer function topological method [6] is used which is based on the transfer function and on the topological properties of the graph associated to the model. In particular, the method defines as unknowns in the identifiability equations not the model parameters (transfer rate constants), but some topological functions of them, i.e. the cycles and paths of the compartmental graph. The mapping of the compartmental parameter space into that of cycles and paths, considerably reduces the number of terms and the number of the high degree terms. Then, this new system of equations is handled by using the Buchberger algorithm [9] which calculates the Gröbner basis. By substituting the model parameters to the cycles and paths in this basis and applying a second time the Buchberger algorithm, it is possible to determine, for each model parameter, if there is one, more than one (and how many), or an infinite number of solutions. The domain of validity of the algorithm is essentially due to the limits of applicability of the Buchberger algorithm in solving the identifiability equations and is thus difficult to be established rigorously. In particular, the Buchberger algorithm has limitations on the number of simultaneous equations to solve, number of unknowns, number of terms and on the nonlinearity degree. Note that the choice of the order relation plays an important role in the computational complexity of the algorithm. We thus checked the algorithm in testing a priori identifiability of the models present in the biomedical literature. From our experience, we can state that, for the first time, it is possible to test automatically a priori structural global identifiability of linear compartmental models characterized by the most general muti input-multi output experimental configuration and by a structure of relatively large connectivity and dimension, i.e. up to twelve compartments.The algorithm described has been implemented in the software GLOBI (GLOBI for GLOBal Identifiability) which is written in PASCAL 6.0 and REDUCE 3.5 [11], where a Gröbner basis package is available with the Buchberger algorithm. In particular, for the user graphical interface, GLOBI utilizes DELPHI [12], an object-oriented program allowing WINDOWS application to PASCAL programs.
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