Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Parameter Identifiability of Linear-Compartmental Mammillary Models

Katherine Clemens, Jonathan Martinez, Anne Shiu, Michaela Thompson, Benjamin Warren

TL;DR

The work analyzes parameter identifiability in linear mammillary models with one input and one output, leveraging input-output equations and a combinatorial coefficient formula to classify which edge parameters are globally identifiable versus SLING for infinite families. It confirms that four of the five mammillary families have all parameters locally identifiable with SLING or global identifiability for key edges, while the Mn$(n)(2,3)$ family exhibits unidentifiability for several edges and is conjectured to extend this unidentifiability pattern. The approach hinges on symmetry arguments, the coefficient map, and identifiable functions to derive explicit identifiability results and formulas relating edge parameters to input-output coefficients. These results illuminate parameter recoverability in structured pharmacokinetic and biological models and suggest directions for future work on more general networks and leaks.

Abstract

Linear compartmental models are a widely used tool for analyzing systems arising in biology, medicine, and more. In such settings, it is essential to know whether model parameters can be recovered from experimental data. This is the identifiability problem. For a class of linear compartmental models with one input and one output, namely, those for which the underlying graph is a bidirected tree, Bortner et al. completely characterized which such models are structurally identifiability, which means that every parameter is generically locally identifiable. Here, we delve deeper, by examining which individual parameters are locally versus globally identifiable. Specifically, we analyze mammillary models, which consist of one central compartment which is connected to all other (peripheral) compartments. For these models, which fall into five infinite families, we determine which individual parameters are locally versus globally identifiable, and we give formulas for some of the globally identifiable parameters in terms of the coefficients of input-output equations. Our proofs rely on a combinatorial formula due to Bortner et al. for these coefficients.

Parameter Identifiability of Linear-Compartmental Mammillary Models

TL;DR

The work analyzes parameter identifiability in linear mammillary models with one input and one output, leveraging input-output equations and a combinatorial coefficient formula to classify which edge parameters are globally identifiable versus SLING for infinite families. It confirms that four of the five mammillary families have all parameters locally identifiable with SLING or global identifiability for key edges, while the Mn family exhibits unidentifiability for several edges and is conjectured to extend this unidentifiability pattern. The approach hinges on symmetry arguments, the coefficient map, and identifiable functions to derive explicit identifiability results and formulas relating edge parameters to input-output coefficients. These results illuminate parameter recoverability in structured pharmacokinetic and biological models and suggest directions for future work on more general networks and leaks.

Abstract

Linear compartmental models are a widely used tool for analyzing systems arising in biology, medicine, and more. In such settings, it is essential to know whether model parameters can be recovered from experimental data. This is the identifiability problem. For a class of linear compartmental models with one input and one output, namely, those for which the underlying graph is a bidirected tree, Bortner et al. completely characterized which such models are structurally identifiability, which means that every parameter is generically locally identifiable. Here, we delve deeper, by examining which individual parameters are locally versus globally identifiable. Specifically, we analyze mammillary models, which consist of one central compartment which is connected to all other (peripheral) compartments. For these models, which fall into five infinite families, we determine which individual parameters are locally versus globally identifiable, and we give formulas for some of the globally identifiable parameters in terms of the coefficients of input-output equations. Our proofs rely on a combinatorial formula due to Bortner et al. for these coefficients.

Paper Structure

This paper contains 16 sections, 14 theorems, 43 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Linear compartmental models
  4. Input-output equations
  5. Identifiability of models
  6. Identifiability of parameters
  7. Identifiable functions
  8. Symmetric edge parameters
  9. Results
  10. Preliminaries
  11. Identifiability of Mn(1,1)
  12. Identifiability of Mn(1,2)
  13. Identifiability of Mn(2,1)
  14. Identifiability of Mn(2,2)
  15. Identifiability of Mn(2,3)
  16. ...and 1 more sections

Key Result

Proposition 2.7

Let $\mathcal{M} = (G, \textit{In}, \textit{Out}, \textit{Leak})$ be a linear compartmental model with $n$ compartments and at least one input. Let $A$ be the compartmental matrix. For $i\in \textit{Out}$, the following equation is an input-output equation for $\mathcal{M}$: where $\partial I$ denotes the $n{\times}n$ matrix in which each diagonal entry is the differential operator $d/dt$ and all

Figures (12)

  • Figure 1: Summary of main results. All mammillary models (up to symmetry) with one input, one output, and no leaks are depicted. The model with input in compartment-$i$ and output in compartment-$j$ is labeled by $(i,j)$. Identifiability properties of all parameters are shown; generically globally identifiable parameters are indicated by (green) boldface arrows, generically locally (but not globally) identifiable parameters are shown in (blue) plain arrows, and unidentifiable parameters are shown with (red) dashed arrows. These properties are proven or conjectured in this work (see Theorem \ref{['thm:summary']} and Conjecture \ref{['conj:2-3']}). The result for the model (1,1) was already known cobelli-lepschy-romaninjacur.
  • Figure 2: The mammillary model $\mathcal{M} = (G, \textit{In}, \textit{Out}, \textit{Leak})$ with $4$ compartments and $\textit{In}=\{1\}$, $\textit{Out}=\{2\}$, and $\textit{Leak}= \varnothing$.
  • Figure 3: The graph $G$ for the mammillary model in Figure \ref{['fig:mammillary']}.
  • Figure 4: The graph $G_{2}^*$, obtained from the graph $G$ in Figure \ref{['fig:original-graph-G']} by removing the outgoing edge from compartment-2 (the output in Figure \ref{['fig:mammillary']}).
  • Figure 5: Spanning incoming forests of $G$ with 3 edges.
  • ...and 7 more figures

Theorems & Definitions (58)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Example 2.5: Example \ref{['ex:mammillary-compartmental-matrix']}, continued
  • Proposition 2.7: Input-output equations MSE
  • Example 2.8: Example \ref{['ex:mammilary-ODE']}, continued
  • Example 2.9: Example \ref{['ex:mamillary-i-o-eqn']}, continued
  • Definition 2.10
  • Proposition 2.11: Coefficients of input-output equations for linear compartmental models BGMSS
  • ...and 48 more