Identifiable reparametrizations of linear compartment models

Structural identifiability concerns finding which unknown parameters of a model can be quantified from given input-output data. Many linear ODE models, used in systems biology and pharmacokinetics, are unidentifiable, which means that parameters can take on an infinite number of values and yet yield the same input-output data. We use commutative algebra and graph theory to study a particular class of unidentifiable models and find conditions to obtain identifiable scaling reparametrizations of these models. Our main result is that the existence of an identifiable scaling reparametrization is equivalent to the existence of a scaling reparametrization by monomial functions. We provide an algorithm for finding these reparametrizations when they exist and partial results beginning to classify graphs which possess an identifiable scaling reparametrization.

[1]  Xiaohua Xia,et al.  On Identifiability of Nonlinear ODE Models and Applications in Viral Dynamics , 2011, SIAM Rev..

[2]  H P Wynn,et al.  Differential algebra methods for the study of the structural identifiability of rational function state-space models in the biosciences. , 2001, Mathematical biosciences.

[3]  N. Meshkat,et al.  Alternative to Ritt's pseudodivision for finding the input-output equations of multi-output models. , 2012, Mathematical biosciences.

[4]  S. Audoly,et al.  On the identifiability of linear compartmental systems: a revisited transfer function approach based on topological properties , 1983 .

[5]  Seth Sullivant,et al.  Identifying Causal Effects with Computer Algebra , 2010, UAI.

[6]  L. D'Angio,et al.  A new differential algebra algorithm to test identifiability of nonlinear systems with given initial conditions , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[7]  B. Sturmfels.  Gröbner bases and convex polytopes , 1995 .

[8]  David A. Cox,et al.  The topology of toric varieties , 2011 .

[9]  Lennart Ljung,et al.  On global identifiability for arbitrary model parametrizations , 1994, Autom..

[10]  J. Graver,et al.  Graduate studies in mathematics , 1993 .

[11]  H. P. Williams.  THEORY OF LINEAR AND INTEGER PROGRAMMING (Wiley-Interscience Series in Discrete Mathematics and Optimization) , 1989 .

[12]  K. R. Godfrey,et al.  Identifiability and indistinguishability of linear compartmental models , 1990 .

[13]  David K. Smith.  Theory of Linear and Integer Programming , 1987 .

[14]  Nicolette Meshkat,et al.  An algorithm for finding globally identifiable parameter combinations of nonlinear ODE models using Gröbner Bases. , 2009, Mathematical biosciences.

[15]  N. Biggs.  Algebraic Graph Theory: COLOURING PROBLEMS , 1974 .

[16]  R N Gunn,et al.  A procedure for generating locally identifiable reparameterisations of unidentifiable non-linear systems by the similarity transformation approach. , 1998, Mathematical biosciences.

[17]  Amos Ben-Zvi,et al.  Identifiability of Linear Time-Invariant Differential-Algebraic Systems. 2. The Differential-Algebraic Approach , 2004 .

[18]  M J Chappell,et al.  Extensions to a procedure for generating locally identifiable reparameterisations of unidentifiable systems. , 2000, Mathematical biosciences.

[19]  Ghislaine Joly-Blanchard,et al.  Equivalence and identifiability analysis of uncontrolled nonlinear dynamical systems , 2004, Autom..

[20]  Keith R. Godfrey,et al.  Some extensions to the exhaustive-modeling approach to structural identifiability , 1985 .