IDENTIFIABILITY OF LINEAR COMPARTMENTAL TREE MODELS

A foundational question in the theory of linear compartmental models is how to assess whether a model is identifiable – that is, whether parameter values can be inferred from noiseless data – directly from the combinatorics of the model. We completely answer this question for those models (with one input and one output) in which the underlying graph is a bidirectional tree. Such models include two families of models appearing often in biological applications: catenary and mammillary models. Our proofs are enabled by two supporting results, which are interesting in their own right. First, we give the first general formula for the coefficients of input-output equations (certain equations that can be used to determine identifiability). Second, we prove that identifiability is preserved when a model is enlarged in specific ways involving adding a new compartment with a bidirected edge to an existing compartment.

[1]  Elizabeth Gross,et al.  Identifiability of linear compartmental models: The singular locus , 2017, Adv. Appl. Math..

[2]  Fei Wang,et al.  A Review of Multi‐Compartment Infectious Disease Models , 2020, International statistical review = Revue internationale de statistique.

[3]  Seth Sullivant,et al.  Identifiability Results for Several Classes of Linear Compartment Models , 2014, Bulletin of Mathematical Biology.

[4]  Heather A. Harrington,et al.  Joining and decomposing reaction networks , 2018, Journal of Mathematical Biology.

[5]  Seth Sullivant,et al.  Identifiable reparametrizations of linear compartment models , 2013, J. Symb. Comput..

[6]  Johan Karlsson,et al.  Minimal output sets for identifiability. , 2012, Mathematical biosciences.

[7]  J. Wagner,et al.  History of pharmacokinetics. , 1981, Pharmacology & therapeutics.

[8]  Claudio Cobelli,et al.  Identifiability results on some constrained compartmental systems , 1979 .

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

[10]  JASMIJN A. BAAIJENS,et al.  On the Existence of Identifiable Reparametrizations for Linear Compartment Models , 2015, SIAM J. Appl. Math..

[11]  Anne Shiu,et al.  Identifiability of Linear Compartmental Models: The Impact of Removing Leaks and Edges , 2021 .

[12]  Anne Shiu,et al.  Identifiability of linear compartmental models: the effect of moving inputs, outputs, and leaks , 2019 .

[13]  L. Childs,et al.  An introduction to compartmental modeling for the budding infectious disease modeler , 2018 .

[14]  Heather A. Harrington,et al.  Linear Compartmental Models: Input-Output Equations and Operations That Preserve Identifiability , 2018, SIAM J. Appl. Math..

[15]  John G. Taylor,et al.  Compartmental-model response function for dendritic trees , 1993, Biological Cybernetics.

[16]  A. W. Kelman,et al.  Compartmental models and their application , 1985 .

[17]  N. Meshkat,et al.  Identifiable Paths and Cycles in Linear Compartmental Models , 2020, Bulletin of Mathematical Biology.

[18]  R. Mulholland,et al.  Analysis of linear compartment models for ecosystems. , 1974, Journal of theoretical biology.

[19]  Arild Thowsen,et al.  Structural identifiability , 1977, 1977 IEEE Conference on Decision and Control including the 16th Symposium on Adaptive Processes and A Special Symposium on Fuzzy Set Theory and Applications.

[20]  Mark S. Cohen,et al.  Chronic exposure to Mn inhalation may have lasting effects: A physiologically-based toxicokinetic model in rats , 2010 .

[21]  I. Karsai,et al.  Compartmental Models of Migratory Dynamics , 2011 .

[22]  Soraya Dhillon,et al.  Basic pharmacokinetics , 2006 .

[23]  N. Chau.  Linear n-compartment catenary models: Formulas to describe tracer amount in any compartment and identification of parameters from a concentration-time curve , 1985 .

[24]  Frédéric Hamelin,et al.  A graph theoretical approach to the parameters identifiability characterisation , 2014, Int. J. Control.

[25]  Joseph T. DiPiro,et al.  Concepts in Clinical Pharmacokinetics , 1996 .

[26]  S. Vajda,et al.  Analysis of unique structural identifiability via submodels , 1984 .

[27]  A. Barron,et al.  A Quantitative Model of Honey Bee Colony Population Dynamics , 2011, PloS one.

[28]  S. Audoly,et al.  Results and Conjectures on the Global Identifiability of Linear Systems , 1985 .

[29]  N. Chau.  Parameter identification in N-compartment mamillary models , 1985 .

[30]  T. Tozer.  Concepts basic to pharmacokinetics. , 1981, Pharmacology & therapeutics.

[31]  Jacques Delforge,et al.  On local identifiability of linear systems , 1984 .

[32]  Gleb Pogudin,et al.  Input-output equations and identifiability of linear ODE models , 2022, IEEE Transactions on Automatic Control.