Parameter identifiability of linear-compartmental mammillary models: methods and sensitivity checks

Pieter Verhoeven; Kenji Takahashi; Andrés Silva; Priya Mukherjee

doi:10.58301/qh8e6525

Home Journals Meridian Journal of Mathematical SciencesVolume 10, Issue 310.58301/qh8e6525

Meridian Journal of Mathematical Sciences

Article14 August 2024

Parameter identifiability of linear-compartmental mammillary models: methods and sensitivity checks

Pieter Verhoeven^1†*, Kenji Takahashi^2†, Andrés Silva³, Priya Mukherjee⁴

¹Faculty of Education, Jagiellon Polytechnic, Poland

²School of Environmental Sciences, Universidad de los Andes del Sur, Chile

³School of Public Health, Nordvik University, Norway

⁴Department of History, Kestrel Institute of Technology, Canada

† These authors contributed equally to this work.

* Corresponding author.

Meridian J. Math. Sci. 2024, 10(3), 90-112; https://doi.org/10.58301/qh8e6525

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.

Keywords:Keywords Linear compartmental modelStructural identifiabilityMammillary modelBidirected-tree modelInput-output equationElementary symmetric polynomialMathematics Subject Classification 93B3092C4537N2534A3034A55

Introduction

Compartmental models are used frequently in applications ranging from ecology (Mulholland and Keener 1974) to disease modeling (Tang et al. 2020) and pharmacology (Cherruault and Sarin 1992). A well-known example arises in pharmacokinetics, in which linear compartmental models are widely used to predict the concentration of an injected drug in the body over time; such knowledge is essential for determining optimal drug dosage and timing. Linear compartmental models additionally have been used to describe the distribution of potassium between plasma and red blood cells, the distribution and turnover of cholesterol in the human body, and the flow of energy within ecosystems (Jacquez 1972).

In such applications, it is crucial to know whether the parameters of models can be uniquely determined from data (Massonis et al. 2021; Muñoz-Tamayo et al. 2018; Stigter 2024; Wieland et al. 2021). This is the problem of parameter identifiability. Our focus here is on structural identifiability, which assumes the ideal situation of noiseless data. Access to noiseless data is generally not realistic, but assessing structural identifiability is nevertheless of great importance, as it is a prerequisite for practical identifiability, which allows for data with noise.

The inspiration for our work is a recent classification of (generic) local identifiability for a family of linear compartmental models, namely, those in which the underlying graph is a bidirected tree. This classification is due to Bortner et al. (2023), and it states the following: A bidirected-tree model with one input and one output is generically locally identifiable if and only if the model has at most 1 leak, and the distance from the input to output is at most 1. In other words, this result tells us exactly which models have all parameters (at least) generically locally identifiable. Such models are called identifiable.

Bortner et al.’s result on bidirected-tree models motivates the following questions:

(Q1): In identifiable bidirected-tree models, which parameters are generically globally identifiable (that is, uniquely identifiable)?
(Q2): In unidentifiable bidirected-tree models, what is the source of this unidentifiability, that is, which parameters are unidentifiable?

The significance of question (Q1) in applications comes from the fact that being able to estimate parameters uniquely is preferable to being able to do so only up to a finite set. Similarly, regarding question (Q2), being able to estimate some but not all parameters may suffice in some applications, so we wish to know which ones can or can not be recovered.

Our work addresses questions (Q1) and (Q2) for an infinite family of bidirected-tree models. These models are mammillary models, which means that the underlying graph is a star graph, consisting of a central compartment connected to a number of peripheral compartments. Mammillary models are of “considerable practical importance” (Godfrey 1983, pg. 31), and instances of mammillary models in biological applications appear in Cherruault and Sarin (1992); Godfrey (1983); Jacquez (1972); and Matis et al. (1976); Nakashima and Benet (1989).

Five (families of) mammillary models are depicted in Figure 1, which summarizes our main results. These results concern all mammillary models with one input, one output, and no leaks. Specifically, we prove the identifiability properties – generically globally identifiable versus generically locally identifiable versus unidentifiable – of most of the edge parameters, as shown in Figure 1 (Theorem 3.2). There are, however, several edges whose identifiability properties we verified computationally for several values of n (here, n denotes the number of compartments), but currently can not prove for all n. These edges are the edges in model (2, 3) that are marked as unidentifiable (indicated by dashed arrows) in Figure 1 (Conjecture 3.18).

**Fig. 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 3.2 and Conjecture 3.18). The result for the model (1,1) was already known (Cobelli et al. 1979, §4.1)

In summary, for the models in Figure 1, we completely answer question (Q1) and we conjecture an answer to question (Q2). We emphasize that, to our knowledge, we are essentially the first to investigate questions (Q1) and (Q2) for infinite families of models. (Chau investigated similar questions for reparametrized versions of mammillary models with leaks at all compartments (Chau 1985b) and so-called catenary models (Chau 1985a); while Cobelli, Lepschy, and Romanin Jacur investigated the mammillary models we label by (1,1) and other models (Cobelli et al. 1979) – see also (Cobelli et al. 2002, §5.8) – but their focus was slightly different.) On the other hand, for models arising in biological applications (cellular signaling, epidemiology, physiology, and other areas), Barreiro and Villaverde recently investigated the occurrence (and origins of) parameters that are generically locally – but not globally – identifiable (so-called SLING parameters, short for “structurally locally identifiable but not globally”) (Barreiro and Villaverde 2023).

Our proofs, like those in several prior works (Ahmed et al. 2024; Bortner et al. 2024; Bortner et al. 2023; Bortner and Meshkat 2022, 2024; Chan et al. 2021; Gerberding et al. 2020; Gross et al. 2019, 2022), involve analyzing input-output equations, which are equations involving parameters, input variables, output variables, and their derivatives. This technique comes from following the differential-algebra approach to identifiability. We also rely on a recent combinatorial formula for coefficients of input-output equations (Bortner et al. 2023), which allows us to analyze whether (and, when possible, how) these coefficients can be used to recover some or all of the parameters.

This article is structured as follows. Section 2 provides background on linear compartmental models and their identifiability. Our results are proven in Section 3, and we conclude with a discussion in Section 4.

Background

In this section, we introduce linear compartmental models (Section 2.1) and their input-output equations (Section 2.2). Subsequently, we define identifiability of models (Section 2.3) and of individual parameters (Section 2.4). Our notation matches that of prior work (Gerberding et al. 2020; Meshkat et al. 2015).

Linear compartmental models

Informally, a linear compartmental model is a directed graph, with certain vertices (compartments) marked as “inputs” (representing inflows into the system), “outputs” (compartments at which experimental measurements can be taken), and “leaks” (outflows from the system). The formal definition follows.

Definition 2.1

A linear compartmental model, denoted by $(G, In, Out, Leak)$ , consists of a directed graph $G = (V, E)$ , in which each vertex $i \in V$ represents a compartment of the model, together with three sets $In, Out, Leak \subseteq V$ which denote the sets of input, output, and leak compartments, respectively.

In a linear compartmental model $(G, In, Out, Leak)$ , a directed edge $j \to i$ in G represents a flow from compartment-j to compartment-i. Each such edge has an associated parameter $k_{ij}$ . Similarly, each leak compartment $p \in Leak$ has an associated parameter, $k_{0 p}$ . However, in this work, we focus on models without leaks (that is, $Leak = \emptyset$ ).

In figures, we represent a linear compartmental model by its graph G, together with arrows for the inputs and the leaks, plus the following symbol for outputs: [Image not available]. See, for instance, Figures 1 and 2. The models depicted in both figures belong to an important class of linear compartmental models, called “mammillary models”, which is the focus of this work.

**Fig. 2** The mammillary model $M = (G, In, Out, Leak)$ with 4 compartments and $In = {1}$ , $Out = {2}$ , and $Leak = \emptyset$

Definition 2.2

Let

M = (G, In, Out, Leak)

be a linear compartmental model.

$M$ is a mammillary model if $G = (V, E)$ is a bidirected-star graph: $V = {1, 2, \dots, n}$ , where $n \geq 3$ , and $E = {1 ⇆ 2, 1 ⇆ 3, \dots, 1 ⇆ n}$ .
$M$ is strongly connected if G is strongly connected, that is, for each unordered pair of vertices i, j in G, there is a directed path in G from i to j and also a directed path from j to i.

Every mammillary model is strongly connected.

Definition 2.3

Let

M = (G, In, Out, Leak)

be a linear compartmental model with n compartments, where

G = (V, E)

. The compartmental matrix of

M

is the

n \times n

matrix

A = (a_{ij})

with entries as follows:

\begin{matrix} a_{ij} : = \{\begin{matrix} - k_{0 i} - \sum_{{p ∣ (i, p) \in E}} k_{pi} & if i = j and i \in Leak, \\ - \sum_{{p ∣ (i, p) \in E}} k_{pi} & if i = j and i \notin Leak, \\ k_{ij} & if i \neq j and j \to i is an edge of G, \\ 0 & if i \neq j and j \to i is not an edge of G . \end{matrix}) . \end{matrix}

Example 2.4

The mammillary model in Figure 2 has the following compartmental matrix:

\begin{matrix} A = [\begin{matrix} - k_{21} - k_{31} - k_{41} & k_{12} & k_{13} & k_{14} \\ k_{21} & - k_{12} & 0 & 0 \\ k_{31} & 0 & - k_{13} & 0 \\ k_{41} & 0 & 0 & - k_{14} \end{matrix}] . \end{matrix}

1

Next, we describe how to use the compartmental matrix to write the ordinary differential equation (ODE) system arising from a model

M = (G, In, Out, Leak)

. Let n denote the number of compartments, and let

x (t) = {(x_{1} (t), x_{2} (t), \dots, x_{n} (t))}^{T}

denote the vector of concentrations of all compartments at time t. For

i \in In

, let

u_{i} (t)

denote the inflow rate into compartment-i at time t. For

i \in Out

, let

y_{i} (t)

denote the corresponding output variable (measurement) at time t. Now

M

defines the following ODE system with inputs, where A is the compartmental matrix of

M

:

\begin{matrix} x^{'} (t) & = A x (t) + u (t), \\ y_{i} (t) & = x_{i} (t), for all i \in Out, \end{matrix}

2

where

u_{i} (t) : = 0

for all

i \notin In

.

Example 2.5

(Example 2.4, continued) Using the compartmental matrix in (1), the ODE system (2) for the model in Figure 2 is as follows:

\begin{matrix} x_{1}^{'} (t) & = (- k_{21} - k_{31} - k_{41}) x_{1} + k_{12} x_{2} + k_{13} x_{3} + k_{14} x_{4} + u_{1} (t) \\ x_{2}^{'} (t) & = k_{21} x_{1} - k_{12} x_{2} \\ x_{3}^{'} (t) & = k_{31} x_{1} - k_{13} x_{3} \\ x_{4}^{'} (t) & = k_{41} x_{1} - k_{14} x_{4} \\ y_{2} (t) & = x_{2} (t) . \end{matrix}

Input-output equations

An input-output equation is an equation that holds along every solution x(t) of the ODEs (2) and involves only the parameters $k_{ij}$ , input variables $u_{ℓ}$ , output variables $y_{m}$ , and their derivatives. The next result is a formula for input-output equations, which is due to (Meshkat et al. 2015, Theorem 2).

Notation 2.6

We let $B^{j, i}$ denote the submatrix obtained from a matrix B by removing row-j and column-i.

Proposition 2.7

(Input-output equations (Meshkat et al. 2015)) Let

M = (G, In, Out, Leak)

be a linear compartmental model with n compartments and at least one input. Let A be the compartmental matrix. For

i \in Out

, the following equation is an input-output equation for

M

:

\begin{matrix} det (\partial I - A) y_{i} = \sum_{j \in I n} {(- 1)}^{i + j} det [{(\partial I - A)}^{j, i}] u_{j}, \end{matrix}

3

where

\partial I

denotes the

n \times n

matrix in which each diagonal entry is the differential operator d/dt and all off-diagonal entries are 0.

Example 2.8

(Example 2.5, continued) Recall that the model in Figure 2 has

In = {1}

and

Out = {2}

. We use the compartmental matrix in (1) to obtain the input-output equation (3):

\begin{matrix} det ([\begin{matrix} d / d t + k_{21} + k_{31} + k_{41} & - k_{12} & - k_{13} & - k_{14} \\ - k_{21} & d / d t + k_{12} & 0 & 0 \\ - k_{31} & 0 & d / d t + k_{13} & 0 \\ - k_{41} & 0 & 0 & d / d t + k_{14} \end{matrix}]) y_{2} \\ = {(- 1)}^{2 + 1} det ([\begin{matrix} - k_{21} & 0 & 0 \\ - k_{31} & d / d t + k_{13} & 0 \\ - k_{41} & 0 & d / d t + k_{14} \end{matrix}]) u_{1}, \end{matrix}

which expands as follows:

\begin{matrix} y_{2}^{(4)} & + (k_{12} + k_{13} + k_{14} + k_{21} + k_{31} + k_{41}) y_{2}^{(3)} \\ + (k_{12} k_{13} + k_{12} k_{14} + k_{12} k_{31} + k_{12} k_{41} + k_{13} k_{14} + k_{13} k_{21} + k_{13} k_{41} \\ + k_{14} k_{21} + k_{14} k_{31}) y_{2}^{(2)} \\ + (k_{12} k_{13} k_{14} + k_{12} k_{13} k_{41} + k_{12} k_{14} k_{31} + k_{13} k_{14} k_{21}) y_{2}^{(1)} \\ = k_{21} u_{1}^{(2)} + (k_{13} k_{21} + k_{14} k_{21}) u_{1}^{(1)} + k_{13} k_{14} k_{21} u_{1} \end{matrix}

4

Next, we introduce notation for an index set for all parameters of a model

M = (G, In, Out, Leak)

, where

G = (V, E)

:

\begin{matrix} P_{M} : = {(j, i) ∣ (j, i) \in E, or i = 0 and j \in Leak} . \end{matrix}

5

We use this set, which has size

| E | + | Leak |

, together with the input-output equation(s) (3), to define the coefficient map of

M

:

\begin{matrix} C : R^{| E | + | Leak |} \to R^{m}, \end{matrix}

which evaluates each vector of parameters

{(k_{ij})}_{(j, i) \in P_{M}}

at the vector of non-constant coefficients of the input-output equation(s). (Here, m denotes the number of such coefficients.)

Example 2.9

(Example 2.8, continued) For the model in Figure 2, we use the input-output equation (4) to obtain the coefficient map

C : R^{6} \to R^{6}

, where

\begin{matrix} C_{1} & : = k_{12} + k_{13} + k_{14} + k_{21} + k_{31} + k_{41} \\ C_{2} & : = k_{12} k_{13} + k_{12} k_{14} + k_{12} k_{31} + k_{12} k_{41} + k_{13} k_{14} \\ + k_{13} k_{21} + k_{13} k_{41} + k_{14} k_{21} + k_{14} k_{31} \\ C_{3} & : = k_{12} k_{13} k_{14} + k_{12} k_{13} k_{41} + k_{12} k_{14} k_{31} + k_{13} k_{14} k_{21} \\ C_{4} & : = k_{21} \\ C_{5} & : = k_{13} k_{21} + k_{14} k_{21} \\ C_{6} & : = k_{13} k_{14} k_{21} . \end{matrix}

6

The next result, Proposition 2.11 below, is used to analyze coefficient maps. We first need the following definition.

Definition 2.10

Let H be a directed graph.

A spanning subgraph is a subgraph of H that includes all vertices of H.
A spanning incoming forest of H is a spanning subgraph such that:
- each node has at most one outgoing edge, and
- the underlying undirected multigraph is a forest, that is, has no cycles.

Bortner et al. gave a combinatorial formula, which is in terms of spanning incoming forests, for the coefficients of input-output equations (Bortner et al. 2023, Theorem 3.1). We state the version of their result for the case of models with one input, one output, and no leaks (which is the situation we focus on in this work), as follows.

Proposition 2.11

(Coefficients of input-output equations for linear compartmental models (Bortner et al. 2023)) Let

n \geq 3

. Let

M = (G, In, Out, Leak)

be an n-compartment model with one input, one output, and no leaks:

In = {j}

,

Out = {i}

, and

Leak = \emptyset

. Write the input-output equation (3) as follows:

\begin{matrix} y_{i}^{(n)} + c_{n - 1} y_{i}^{(n - 1)} + \dots + c_{1} y_{i}^{'} + c_{0} y_{i} = d_{n - 1} u_{j}^{(n - 1)} + \dots + d_{1} u_{j}^{'} + d_{0} u_{j} . \end{matrix}

7

Then the coefficients of the input-output equation (7) are given by:

\begin{matrix} c_{k} & = \sum_{F \in F_{n - k} (G)} π_{F} for k = 0, 1, \dots, n - 1, and \\ d_{k} & = \sum_{F \in F_{n - k - 1}^{j, i} (G_{i}^{*})} π_{F} for k = 0, 1, \dots, n - 1, \end{matrix}

8

where:

$G_{i}^{*}$ is the directed graph obtained from G by removing all outgoing edges from vertex-i (the output),
$F_{ℓ} (G)$ is the set of all spanning incoming forests of G with exactly $ℓ$ edges,
$F_{ℓ}^{j, i} (G_{i}^{*})$ is the set of all spanning incoming forests of $G_{i}^{*}$ with exactly $ℓ$ edges, such that some connected component (of the underlying undirected graph) contains both of the vertices j and i,
$π_{F}$ is the product of edge labels of a graph F, that is, $π_{F} : = \prod_{e \in E_{F}} L (e)$ , where L(e) is the label of edge e, and $E_{F}$ is set of edges of F. If $E_{F} = \emptyset$ , then $π_{F} : = 1$ .

Remark 2.12

( $c_{0} = d_{n - 1} = 0$ ) In the context of Proposition 2.11, we have $c_{0} = 0$ . To see this, observe that the formula for $c_{0}$ in (8) is a sum over certain n-edge cycle-free subgraphs of G. However, G has only n vertices, and hence has no n-edge cycle-free subgraphs. We conclude that $c_{0}$ is the empty sum and hence is 0. An alternate proof can be given directly from (3). Similarly, $d_{n - 1} = 0$ .

Remark 2.13

In Proposition 2.11, the input-output equation (7) corrects a sign error in (Bortner et al. 2023, Theorem 3.1), which erroneously contains a factor of ${(- 1)}^{i + j}$ on the right-hand side.

Example 2.14

(Example 2.9, continued) Recall that the 4-compartment mammillary model $M = (G, In, Out, Leak)$ in Figure 2 has $In = {1}$ , $Out = {2}$ , and $Leak = \emptyset$ . Following the notation in (7), the coefficients $c_{i}$ and $d_{i}$ are given by $(c_{3}, c_{2}, c_{1}, d_{2}, d_{1}, d_{0}) : = (C_{1}, C_{2}, C_{3}, C_{4}, C_{5}, C_{6})$ , where the $C_{i}$ are as in (6), and also $c_{0} = 0$ . Next, the graphs G and $G_{2}^{*}$ are shown in Figures 3 and 4, respectively. It is now straightforward to check that the coefficients $c_{i}$ and $d_{i}$ satisfy the formulas (8). For instance, $c_{1} = C_{3}$ is a sum over the 4 spanning incoming forests of G with $4 - 1 = 3$ edges; these 4 forests are depicted in Figure 5. Similarly, $d_{1} = C_{5}$ is a sum over the 2 spanning incoming forests of $G_{2}^{*}$ with $4 - 1 - 1 = 2$ edges in which compartments 1 and 2 are connected; these 2 forests are shown in Figure 6.

**Fig. 3** The graph G for the mammillary model in Figure 2

**Fig. 4** The graph $G_{2}^{*}$ , obtained from the graph G in Figure 3 by removing the outgoing edge from compartment-2 (the output in Figure 2)

**Fig. 5** Spanning incoming forests of G with 3 edges.

**Fig. 6** Spanning incoming forests of $G_{2}^{*}$ with 2 edges in which compartments 1 and 2 are connected

Identifiability of models

The idea behind the problem of model identifiability is as follows: Given measured values of how much enters and exits the input and output compartments, respectively, can we infer all parameter values of a model? To be more precise, we are asking whether, from generic values of the inputs and initial conditions, we can recover the parameters (at least locally, which means up to a finite set) from exact measurements of the inputs and the outputs. This is the concept of generic local identifiability.

This notion of identifiability is equivalent (for strongly connected models with at least one input) to Definition 2.15 below, which checks identifiability by way of coefficient maps. This equivalence was proven by (Ovchinnikov et al. 2023, Main Result 3).

In what follows, we use “almost every” in the usual measure-theoretic sense: “almost every x in a set X” means “for all x in $X \ Z$ , for some measure-zero subset Z of X”.

Definition 2.15

(Identifiability of

M

) Let

M = (G, In, Out, Leak)

be a linear compartmental model that is strongly connected and has at least one input. Let

G = (V, E)

, and let

C : R^{| E | + | Leak |} \to R^{m}

denote the coefficient map of

M

.

$M$ is generically locally identifiable if almost every point in $R^{| E | + | Leak |}$ has an open neighborhood on which $C$ is injective.
$M$ is unidentifiable if $M$ is not generically locally identifiable.

The following result, which was proved by (Bortner et al. 2023, Corollary 5.4), characterizes identifiability of mammillary models (Definition 2.2(1)) with one input and one output.

Proposition 2.16

(Identifiability of mammillary models) A mammillary model $M = (G, In, Out, Leak)$ with $| In | = | Out | = 1$ is generically locally identifiable if and only if $| L e a k | \leq 1$ and one or more of the following hold: (1) $I n = O u t$ , (2) $I n = {1}$ , or (3) $O u t = {1}$ .

Example 2.17

(Example 2.14, continued) For the mammillary model in Figure 2, we have $I n = {1}$ , $Out = {2}$ , and no leaks. Therefore, by Proposition 2.16, the model is generically locally identifiable.

In the next subsection, we examine whether the individual parameters of this model are generically globally identifiable versus generically locally identifiable.

Identifiability of parameters

Definition 2.15 above concerns the identifiability of models, whereas the next definition pertains to the identifiability of individual parameters of a model (cf. Hong et al. 2018, Definition 2.5).

Definition 2.18

(Identifiability of parameter $k_{pq}$ ) Let $M = (G, In, Out, Leak)$ be a strongly connected, linear compartmental model with at least one input. Let $G = (V, E)$ , and let $C : R^{| E | + | Leak |} \to R^{m}$ denote the coefficient map of $M$ . Let $k_{pq}$ be a parameter of $M$ , i.e., (q, p) is in the index set $P_{M}$ , in (5). Write each parameter vector ${(k_{ij})}_{(j, i) \in P_{M}}$ as $(k_{pq} ; \tilde{k})$ , where $\tilde{k} : = {(k_{ij})}_{(j, i) \in P_{M} \ {(q, p)}}$ ; this allows us to write the coefficient map as $C = C (k_{pq} ; \tilde{k})$ .

The parameter

k_{pq}

is:

globally identifiable if, for every $(k_{pq}^{*} ; {\tilde{k}}^{*})$ in $R^{| E | + | Leak |}$ , the following set (which is a projection of a preimage of $C (k_{pq}^{*} ; {\tilde{k}}^{*})$ ) has size one:
$\begin{matrix} {k_{pq} ∣ there exists \tilde{k} \in R^{| E | + | Leak |} such that C (k_{pq} ; \tilde{k}) = C (k_{pq}^{*} ; {\tilde{k}}^{*})} ; \end{matrix}$ 9
generically globally identifiable if, for almost every $(k_{pq}^{*} ; {\tilde{k}}^{*})$ in $R^{| E | + | Leak |}$ , the set (9) has size one;
generically locally identifiable if, for almost every $(k_{pq}^{*} ; {\tilde{k}}^{*})$ in $R^{| E | + | Leak |}$ , the set (9) is finite; and
unidentifiable if $k_{pq}$ is not generically locally identifiable.

Notation 2.19

Following Barreiro and Villaverde, we use SLING (which stands for “structurally locally identifiable but not globally”) to refer to a parameter that is generically locally identifiable, but not generically globally identifiable (Barreiro and Villaverde 2023).

Remark 2.20

(Identifiability of models versus parameters) A model is generically locally identifiable (Definition 2.15(1)) if and only if all of its parameters are generically locally identifiable (Definition 2.18(3)); cf. (Hong et al. 2018, Remark 2.6(a)). Equivalently, a model is unidentifiable if and only if at least one of its parameters is unidentifiable.

Remark 2.21

(Software for checking identifiability) Identifiability of parameters can be checked using software. One option is the Maple software SIAN (Structural Identifiability ANalyser) (Hong et al. 2019), which is based on theory developed by Hong et al. (2018). (This theory is developed over $C$ , whereas we work over $R$ ; hence, users of SIAN need to be careful when interpreting the output.) Another option is the Julia package StructuralIdentifiability.jl, which implements theory of Dong et al. (2023).

Remark 2.22

(Restricting parameter values) A meaningful extension of Definition 2.18 would be to include, as part of the setup, a subset of the full parameter space $R^{| E | + | L e a k |}$ (or a collection of subsets of $R$ , one for each parameter). The reason would be to allow a researcher to take into account known information about parameter values (positivity, for instance, or restriction to an interval). Here, however, for simplicity, we do not incorporate this addition.

Identifiable functions

One of our primary methods for proving that certain parameters are generically globally identifiable or generically locally identifiable is through the theory of “identifiable functions” (see Definition 2.23 and Proposition 2.25 below). This approach makes precise the idea that identifiable parameters are the ones that we can “solve for” in terms of the coordinates of the coefficient map.

We follow the notation of Meshkat et al. (2018) in what follows.

Definition 2.23

Let

C : R^{| E | + | Leak |} \to R^{m}

be the coefficient map of a linear compartmental model, and let

f : R^{| E | + | Leak |} \to R

be some other function. The function f is:

an identifiable function from $C$ if, for all $p, p^{'} \in R^{| E | + | Leak |}$ , we have that $C (p) = C (p^{'})$ implies that $f (p) = f (p^{'})$ .
a generically identifiable function from $C$ if f is identifiable from $C$ on some open, dense subset $U \subseteq R^{| E | + | Leak |}$ .
a locally identifiable function from $C$ if there is an open, dense subset $U \subseteq R^{| E | + | Leak |}$ such that, for all $p \in U$ , there is an open neighborhood $U_{p}$ of p such that f is identifiable from $C$ on $U_{p}$ .

Remark 2.24

(Identifiable parameters and functions) The relationship between Definitions 2.18 and 2.23 is as follows. For a parameter $k_{ij}$ of a model with coefficient map $C$ , if the function $f = k_{ij}$ is identifiable (respectively, generically identifiable or locally identifiable), then the parameter $k_{ij}$ is globally identifiable (respectively, generically globally identifiable or generically locally identifiable).

The following proposition, which is similar in spirit to (Meshkat et al. 2018, Proposition 4.4) (and can be viewed as an instantiation of the ideas in (Hong et al. 2018, Proposition 3.4) and (Ovchinnikov et al. 2022, Definition 2.5)), will help us prove identifiability properties of parameters:

Proposition 2.25

Let

M = (G, In, Out, Leak)

be a linear compartmental model with coefficient map

C : R^{| E | + | Leak |} \to R^{m}

. Let

k_{ij}

be a parameter of

M

. Suppose that there is a function

g : R \times R^{| E | + | Leak |} \to R

, which we write as

g = g (z ; {(k_{ij})}_{(j, i) \in P_{M}})

, for which:

for some $d \geq 1$ , there exist polynomial functions $q_{0}, q_{1}, \dots, q_{d} : R^{| E | + | Leak |} \to R$ that are identifiable from $C$ , such that
$\begin{matrix} g = q_{d} z^{d} + q_{d - 1} z^{d - 1} + \dots + q_{0}, \end{matrix}$
and the leading coefficient $q_{d}$ is not the zero function; and
the function ${g |}_{z = k_{ij}} : R^{| E | + | Leak |} \to R$ is the zero function.

Then the parameter

k_{ij}

is generically locally identifiable. Moreover, if

d = 1

, then the parameter

k_{ij}

is generically globally identifiable.

Proof

Let $g, q_{0}, q_{1}, \dots, q_{d}$ , and $k_{ij}$ be as in the statement of the proposition. The set $U : = R^{| E | + | Leak |} \ {q_{d} = 0}$ is an open, dense subset of $R^{| E | + | Leak |}$ (here we use the fact that the polynomial $q_{d}$ is nonzero). In particular, the complement of U has measure zero.

Fix $(k_{ij}^{*} ; {\tilde{k}}^{*}) \in U$ (here, the notation $(k_{ij}^{*} ; {\tilde{k}}^{*})$ is like that in Definition 2.18). By Definition 2.18(2), it suffices to show that there are only finitely many values $k_{ij}^{* *}$ for which there exist some ${\tilde{k}}^{* *} \in R^{| E | + | Leak | - 1}$ with $C (k_{ij}^{* *} ; {\tilde{k}}^{* *}) = C (k_{ij}^{*} ; {\tilde{k}}^{*})$ .

To this end, recall that

q_{0}, q_{1}, \dots, q_{d}

are identifiable from

C

. This fact implies that the following holds for all

i = 0, 1, \dots, d

, whenever

C (k_{ij}^{* *} ; {\tilde{k}}^{* *}) = C (k_{ij}^{*} ; {\tilde{k}}^{*})

:

\begin{matrix} q_{i}^{*} & : = q_{i} {|_{(k_{ij} ; \tilde{k}) = (k_{ij}^{*} ; {\tilde{k}}^{*})} = q_{i} |}_{(k_{ij} ; \tilde{k}) = (k_{ij}^{* *} ; {\tilde{k}}^{* *)}} . \end{matrix}

Now recall that

{g |}_{z = k_{ij}}

is the zero function (by assumption). It now follows that, if

C (k_{ij}^{* *} ; {\tilde{k}}^{* *}) = C (k_{ij}^{*} ; {\tilde{k}}^{*})

, then

k_{ij}^{* *}

is a root of the following univariate polynomial (which is nonzero because we constructed U to be disjoint from

{q_{d} = 0}

):

\begin{matrix} {g |}_{(k_{ij} ; \tilde{k}) = (k_{ij}^{*} ; {\tilde{k}}^{*})} = q_{d}^{*} z^{d} + q_{d - 1}^{*} z^{d - 1} + \dots + q_{0}^{*} . \end{matrix}

Thus,

k_{ij}^{* *}

can take at most d values, and so we conclude that the parameter

k_{ij}

is generically locally identifiable. Additionally, if

d = 1

, then there is only one possible value (namely,

k_{ij}^{* *} = k_{ij}^{*}

) and so, in this case, the parameter

k_{ij}

is generically globally identifiable.

□

Example 2.26

(Example 2.17, continued) We return to the model in Figure 2 and its coefficient map (6). We saw that $C_{4} = k_{21}$ , and so the parameter $k_{21}$ is easily recovered from the coefficients $C_{i}$ . More precisely, $k_{21}$ is an identifiable function from $C$ , so Remark 2.24 implies that $k_{21}$ is globally identifiable. Alternatively, we can draw a slightly weaker conclusion by noting that $k_{21}$ satisfies the hypotheses of Proposition 2.25 with $g : = - k_{21} + z$ (so, $q_{0} = - k_{21}$ and $q_{1} = 1$ ), and hence $k_{21}$ is generically globally identifiable. It turns out that no other parameter of this model is generically globally identifiable; in fact, all remaining parameters are SLING. This fact can be proven directly or by examining the symmetry in the model (see Example 2.29 below).

In Example 2.26, we saw that one of the coefficients (namely, $C_{4}$ ) equals the parameter $k_{21}$ . This parameter corresponds to the edge from the input (at compartment-1) to the output (at compartment-2). This observation generalizes, as follows.

Lemma 2.27

(Global identifiability of edge from input to output) Let $M = (G, In, Out, Leak)$ be a strongly connected, linear compartmental model with $In = {i}$ , $Out = {j}$ , and $Leak = \emptyset$ . If $i \to j$ is an edge of G (with parameter $k_{ji}$ ), then $k_{ji}$ is globally identifiable.

Proof

Proposition 2.11 implies that one of the coefficients of the input-output equations, namely, $d_{n - 2}$ , as in (7), equals $k_{ji}$ . Now the result follows from Remark 2.24. $□$

Remark 2.28

Gogishvili constructed a large database of linear compartmental models, and observed that Lemma 2.27 holds for all models in the database (Gogishvili 2024, §5).

Symmetric edge parameters

We end this section by asserting that edges that are symmetric share the same identifiability properties (Lemma 2.32 below). This simple idea has appeared in prior work (e.g., Gogishvili 2024, §2.2 and Cobelli et al. (1979)), and we motivate this result through the following example.

Example 2.29

(Example 2.26, continued) Observe in Figure 2 the symmetry between compartments 3 and 4. This symmetry tells us that the parameters for edges incident to 3 (namely, $k_{13}$ and $k_{31}$ ) and those incident to 4 (namely, $k_{14}$ and $k_{41}$ ) are not generically globally identifiable, because their values are indistinguishable from those of their symmetric pair. This indistinguishability also can be seen in the coefficients (6) of the input-output equation: Each coefficient is invariant under exchanging $k_{13}$ and $k_{14}$ with $k_{31}$ and $k_{41}$ , respectively. We make these ideas precise through the next definition and lemma.

Definition 2.30

Let n be a positive integer.

An automorphism of a directed graph $G = (V, E)$ , where $V = {1, 2, \dots, n}$ , is a permutation $σ$ of V such that $σ (G) : = (V, σ (E)) = G$ , where $σ (E) : = {σ (e) ∣ e \in E}$ and $σ (e) : = (σ (i), σ (j))$ , if $e = (i, j) \in E$ .
For linear compartmental models $M = (G, In, Out, Leak)$ and $M^{'} = (G^{'}, {In}^{'}, {Out}^{'}, {Leak}^{'})$ with $G = G^{'}$ , a morphism from $M$ to $M^{'}$ is an automorphism $σ$ of G such that $σ (In) = {In}^{'}$ , $σ (Out) = {Out}^{'},$ and $σ (Leak) = {Leak}^{'}$ .
An automorphism of a linear compartmental model $M$ is a morphism from $M$ to $M$ .

Remark 2.31

(Input-output equations under morphisms) Given a morphism from $M$ to $M^{'}$ , it is straightforward to see from definitions that the input-output equations (3) for $M^{'}$ are obtained from the input-output equations for $M$ by permuting the subscripts according to $σ$ (i.e., we replace $k_{ij}$ and $k_{0 j}$ by, respectively, $k_{σ (i), σ (j)}$ and $k_{0 σ (j)}$ , for relevant indices i, j).

Lemma 2.32

(Symmetric edge parameters are SLING or unidentifiable) Consider a linear compartmental model $M = (G, In, Out, Leak)$ , where $G = (V, E)$ . Consider two edges, $e = (q, p) \in E$ and $e^{'} = (q^{'}, p^{'}) \in E$ , with $e \neq e^{'}$ . If there exists an automorphism $σ$ of $M$ such that $σ (e) = e^{'}$ , then the parameters for e and $e^{'}$ (namely, $k_{pq}$ and $k_{p^{'} q^{'}}$ ) are either both SLING parameters or both unidentifiable parameters of $M$ .

Proof

Let $M$ be a linear compartmental model, with automorphism $σ$ and edges $e = (q, p)$ and $e^{'} = (q^{'}, p^{'})$ , as in the statement of the lemma. Our first aim is to show that the parameters $k_{pq}$ and $k_{p^{'} q^{'}}$ have the same identifiability properties (as in Definition 2.18). Accordingly, consider any parameter vector $k^{*} \in R^{| E | + | Leak |}$ . It suffices to show that the projected set (9) for the edge $e = (q, p)$ is equal to the corresponding projected set for $e^{'} = (q^{'}, p^{'})$ . In fact, we need only show one containment, as the reverse containment will follow by applying the same argument to the inverse morphism $σ^{- 1}$ .

Let

C : R^{| E | + | Leak |} \to R^{m}

denote the coefficient map of

M

, and let

π_{pq} : R^{| E | + | Leak |} \to R

(respectively,

π_{p^{'} q^{'}}

) denote the projection onto the coordinate

k_{pq}

(respectively,

k_{p^{'} q^{'}}

). Now consider the following diagram, where

\tilde{σ}

permutes the coordinates according to

σ

(the coordinates

k_{ij}

and

k_{0 j}

are sent to, respectively,

k_{σ (i), σ (j)}

and

k_{0 σ (j)}

, for relevant indices i, j):

In this diagram, the upper triangle commutes by construction. The lower triangle also commutes, by Remark 2.31 and the fact that $σ$ is an automorphism.

We claim that the fact that the diagram above commutes, implies that the projected set (9) for e, namely, $π_{pq} (C^{- 1} (C (k^{*})))$ is contained in the projected set for $e^{'}$ , namely, $π_{p^{'} q^{'}} (C^{- 1} (C (k^{*})))$ . Indeed, this is a straightforward “diagram chasing” argument, which we outline as follows. Given $x \in π_{pq} (C^{- 1} (C (k^{*})))$ , let $\bar{y} : = \tilde{σ} (\bar{x})$ , where $π_{pq} (\bar{x}) = x$ and $\bar{x} \in C^{- 1} (C (k^{*}))$ . The lower commutative triangle is used to show that $\bar{y} \in C^{- 1} (C (k^{*}))$ , while the upper commutative triangle is used to show that $π_{p^{'} q^{'}} (\bar{y}) = x$ . Hence, $x \in π_{p^{'} q^{'}} (C^{- 1} (C (k^{*})))$ .

To complete the proof, we need only show that the parameter $k_{pq}$ is not generically globally identifiable. Accordingly, let Z denote the hyperplane in $R^{| E | + | Leak |}$ defined by $k_{pq} = k_{p^{'} q^{'}}$ (so, Z is a measure-zero subset of $R^{| E | + | Leak |}$ ). Let $k^{*} : = {(k_{ij}^{*})}_{(j, i) \in P_{M}} \in R^{| E | + | Leak |} \ Z$ . Consider $\tilde{σ} (k^{*}) = {(k_{σ (i) σ (j)}^{*})}_{(j, i) \in P_{M}}$ , where $σ (0) : = 0$ (to account for leak parameters $k_{0 j}^{*}$ ). From the commutative diagram shown earlier in the proof, we have $C (k^{*}) = C (\tilde{σ} (k^{*}))$ . Hence, the projected set (9) for e, contains both $k_{pq}^{*}$ and $k_{σ (p) σ (q)}^{*}$ (which are distinct, because $k^{*} \notin Z$ ). We conclude that, as desired, $k_{qp}$ is not generically globally identifiable. $□$

Results

In this section, we investigate the identifiability of all parameters in mammillary models with one input, one output, and no leaks. For ease of notation, we let $M_{n} (i, j)$ denote the n-compartment mammillary model $(G, In, Out, Leak)$ with $In = {i}$ , $Out = {j}$ , and $Leak = \emptyset$ . For instance, $M_{4} (1, 2)$ denotes the model shown earlier in Figure 2.

Up to symmetry, the mammillary models

M_{n} (i, j)

fall into five families. These families are defined by the location of the input and the output, so we consider the following representatives, one for each of the five families:

\begin{matrix} M_{n} (1, 1), M_{n} (1, 2), M_{n} (2, 1), M_{n} (2, 2), and M_{n} (2, 3) . \end{matrix}

10

Remark 3.1

Proposition 2.16 implies that, among the models (10), only the model $M_{n} (2, 3)$ is unidentifiable or, equivalently, by Remark 2.20, contains unidentifiable parameters. Hence, every parameter in the remaining four models in (10) is generically locally identifiable (and thus is SLING or generically globally identifiable).

Our main result, which was summarized earlier in Figure 1, is as follows.

Theorem 3.2

(Main result) For mammillary models with one input, one output, and no leaks, the parameters have the following identifiability properties:

For all $n \geq 3$ , every parameter of $M_{n} (1, 1)$ is SLING.
For all $n \geq 4$ , every parameter of $M_{n} (1, 2)$ is SLING, except the parameter $k_{21}$ labeling the edge from the input to output, which is globally identifiable.
For all $n \geq 4$ , every parameter of $M_{n} (2, 1)$ is SLING, except the parameters $k_{12}$ and $k_{21}$ labeling the edges between the input and output, which are, respectively, globally identifiable and generically globally identifiable.
For all $n \geq 4$ , every parameter of $M_{n} (2, 2)$ is SLING, except the parameters $k_{12}$ and $k_{21}$ labeling the edges between the input and output, which are generically globally identifiable.
For all $n \geq 5$ , the parameters of $M_{n} (2, 3)$ that label the edges from non-input, non-output peripheral compartments to the central compartment (namely, $k_{14}, k_{15}, \dots, k_{1 n}$ ) are SLING.

The remainder of this section is dedicated to proving Theorem 3.2 (see Propositions 3.9, 3.11, 3.12, 3.14, and 3.16 below). We also state a conjecture concerning model $M_{n} (2, 3)$ that, if proven, would elevate Theorem 3.2 to match what is depicted in Figure 1 (see Conjecture 3.18). This conjecture is supported by examples we computed using software (see Remark 2.21).

Remark 3.3

In Theorem 3.2, we see each of the models $M_{n} (1, 2)$ and $M_{n} (2, 1)$ has one parameter that is globally identifiable (which is stronger than being generically globally identifiable). This global identifiability comes from the fact that such a parameter corresponds to the edge from input to output (recall Lemma 2.27).

Preliminaries

All five model families (10) share the same compartmental matrix A (for a given number of compartments n). Thus, the set of coefficients on the left-hand side of the input-output equation (3) also is the same. Accordingly, we give a formula for these coefficients in this subsection (Proposition 3.7 below). Appearing in this formula (and other formulas for coefficients we see later in this section) are elementary symmetric polynomials.

Definition 3.4

Let k and n be positive integers, with

1 \leq k \leq n

. Let

X_{n}

denote the set of variables

x_{1}, x_{2}, \dots, x_{n}

. The k-th elementary symmetric polynomial on

X_{n}

, which we denote by

e_{k} (X_{n})

, is given by

\begin{matrix} e_{k} (X_{n}) : = \sum_{\begin{matrix} I \subseteq [n] \\ | I | = k \end{matrix}} (\prod_{i \in I}, x_{i}) . \end{matrix}

We additionally adopt the following standard convention:

e_{0} (X_{n}) : = 1

.

Example 3.5

(Example 2.26, continued) We revisit the mammillary model

M_{4} (1, 2)

in Figure 2. The left-hand side coefficients of the input-output equation (see (6)) can be rewritten in terms of elementary symmetric polynomials, as follows:

\begin{matrix} c_{3} & = C_{1} = e_{1} (I) + k_{21} + k_{31} + k_{41}, \\ c_{2} & = C_{2} = e_{2} (I) + k_{21} e_{1} (I \ {k_{12}}) + k_{31} e_{1} (I \ {k_{13}}) + k_{41} e_{1} (I \ {k_{14}}) \\ c_{1} & = C_{3} = e_{3} (I) + k_{21} e_{2} (I \ {k_{12}}) + k_{31} e_{2} (I \ {k_{13}}) + k_{41} e_{2} (I \ {k_{14}}), \end{matrix}

11

where

I : = {k_{12}, k_{13}, k_{14}}

is the set of all “incoming” edge labels (that is, labels of edges of the graph G in Figure 3 that are directed toward the central compartment).

Example 3.5 displayed formulas (11) for the left-hand side coefficients for $M_{4} (1, 2)$ . We give a new formula for these coefficients, using subgraphs of the graph in Figure 7, as follows.

Example 3.6

(Example 3.5, continued) For the model

M_{4} (1, 2)

, the left-hand side coefficients of the input-output equation (shown in (6) and (11)) can be rewritten by grouping terms based on whether they contain

k_{21}

, contain

k_{12}

, or contain neither

k_{21}

nor

k_{12}

:

\begin{matrix} c_{3} & = k_{21} + k_{12} + g_{1} \\ c_{2} & = k_{21} e_{1} (Σ) + k_{12} g_{1} + g_{2} \\ c_{1} & = k_{21} e_{2} (Σ) + k_{12} g_{2}, \end{matrix}

12

where

\begin{matrix} Σ & : = {k_{13}, k_{14}}, g_{1} : = k_{13} + k_{14} + k_{31} + k_{41}, and \\ g_{2} & : = k_{13} k_{14} + k_{13} k_{41} + k_{14} k_{31} . \end{matrix}

Notice that

g_{1}

is the sum over all edges in the

n = 4

version of the graph in Figure 7, and

g_{2}

is the sum over pairs of edges in Figure 7 that do not form a cycle. Also, it is straightforward to deduce the following equation from the equations (12):

\begin{matrix} k_{12}^{2} c_{3} - k_{12} c_{2} + c_{1} = (k_{12}^{2} - k_{12} e_{1} (Σ) + e_{2} (Σ)) k_{21} + k_{12}^{3} . \end{matrix}

13

Next, alternate formulas for

g_{1}

and

g_{2}

are given by:

Image not available

14

The formulas (12) and equations (13)–(14) generalize to what is shown in (15), (17), and (18), respectively, in the next result.

**Fig. 7** Graph obtained from the star graph in Figure 3 by removing the pair of edges between compartment-1 and compartment-2

Proposition 3.7

(Left-hand side coefficients) Let

n \geq 3

. Consider an n-compartment mammillary model

M = (G, In, Out, Leak)

with

| In | = | Out | = 1

and

Leak = \emptyset

. Let

c_{0}, c_{1}, \dots, c_{n - 1}

denote the coefficients of the left-hand side of the input-output equation, as in (7). Then the following hold:

$c_{0} = 0$ , and the coefficients $c_{1}, c_{2}, \dots, c_{n - 1}$ are given by the following formulas:
$\begin{matrix} c_{n - 1} & = k_{21} + k_{12} g_{0} + g_{1} \\ c_{n - 2} & = k_{21} e_{1} (Σ) + k_{12} g_{1} + g_{2} \\ c_{n - 3} & = k_{21} e_{2} (Σ) + k_{12} g_{2} + g_{3} \\ ⋮ \\ c_{1} & = k_{21} e_{n - 2} (Σ) + k_{12} g_{n - 2} + g_{n - 1}, \end{matrix}$ 15
where $Σ : = {k_{13}, k_{14}, \dots, k_{1 n}}$ and, for $i = 0, 1, 2, \dots, n - 1$ , we denote:
$\begin{matrix} g_{i} : = \sum_{F is an i - edge spanning, incoming forest of \tilde{G}} π_{F}, \end{matrix}$ 16
where $\tilde{G}$ denotes the graph in Figure 7.
The following equation holds:
$\begin{matrix} k_{12}^{n - 2} c_{n - 1} - k_{12}^{n - 3} c_{n - 2} + k_{12}^{n - 4} c_{n - 3} - \dots \pm c_{1} \\ = (k_{12}^{n - 2} - k_{12}^{n - 3} e_{1} (Σ) + k_{12}^{n - 4} e_{2} (Σ) - \dots \pm e_{n - 2} (Σ)) k_{21} + k_{12}^{n - 1} . \end{matrix}$ 17
The terms $g_{i}$ are given by $g_{0} = 1$ , $g_{n - 1} = 0$ , and the following formula:
Image not available
18
where $Σ : = {k_{13}, k_{14}, \dots, k_{1 n}}$ is the set of all “incoming” edge labels of $\tilde{G}$ (the graph in Figure 7), and where M denotes the following $(n - 2) \times (n - 2)$ matrix:
Image not available
19
The determinant of this matrix M is, up to sign, the Vandermonde polynomial $\prod_{3 \leq j < ℓ \leq n} (k_{1 j} - k_{1 ℓ})$ .

Proof

The equality

c_{0} = 0

was given earlier (Remark 2.12). Next, for

i = 1, 2, \dots, n - 1

, recall from Proposition 2.11 that the coefficient

c_{i}

is the sum of all terms

π_{F}

, where F is a spanning incoming forest that is an

(n - i)

-edge subgraph of the star graph G (in Figure 3). Such a forest can not contain both of the edges

k_{21}

and

k_{12}

(to avoid a cycle), so each

c_{i}

is a sum over three types of forests:

$\underline{forests involving k_{21} (but not k_{12})}$ – in this case, as $k_{21}$ is outgoing from compartment-1, the remaining edges must be chosen from the “incoming” edges (those directed toward compartment-1) from $3, 4, \dots, n$ (and can be arbitrarily chosen); these edges are indexed by the set $Σ$ ;
$\underline{forests involving k_{12} (but not k_{21})}$ – in this case, the edge $k_{12}$ can be combined with any set of edges of $\tilde{G}$ that do not contain a cycle;
$\underline{forests involving neither k_{12} nor k_{21}}$ – these are spanning, incoming forests of $\tilde{G}$ .

Viewing each

c_{i}

as such a sum exactly yields the formulas (15), as claimed.

Next, we note that

\begin{matrix} g_{0} = 1 and g_{n - 1} = 0, \end{matrix}

20

because (1) we have

π_{F} = 1

for the subgraph F with no edges, and (2) spanning, incoming forests of

\tilde{G}

have at most

n - 2

edges (and hence none have

n - 1

edges).

Next, it is straightforward to use the equations (15) and (20) to obtain the equation (17). To provide some detail, we use (15) to obtain expressions for $k_{12}^{n - 2} c_{n - 1}$ , $k_{12}^{n - 3} c_{n - 2}$ , ..., $c_{1}$ , and then we observe that the terms involving $g_{i}$ ’s “telescope” as they are added together, so that subsequently what remains of these terms is $k_{12}^{n - 1} g_{0} + g_{n - 1}$ , which by (20), equals $k_{12}^{n - 1}$ .

We next prove the formula for the $g_{i}$ ’s given in (18). Recall that $g_{i}$ is the sum of all terms $π_{F}$ , where F is a spanning incoming forest that is an i-edge subgraph of the star graph $\tilde{G}$ in Figure 7. By definition, such forests contain at most 1 “outgoing” edge (i.e., an edge of the form $1 \to j$ ). Now it is straightforward to check that $e_{i} (Σ)$ is the subsum of $g_{i}$ corresponding to forests with no “outgoing” edges, and the remaining terms on the right-hand side of (18) form the subsum corresponding to forests with exactly 1 “outgoing” edge (a similar idea underlies the enumeration of spanning incoming forests shown in Figure 5).

Finally, the determinant of the matrix M is well known (see equation (6) in the proof of (Gross et al. 2022, Theorem 6.1)). $□$

We end this subsection by recalling a useful fact about elementary symmetric polynomials; this result is well known (cf. Gross et al. 2022, Proof of Theorem 6.3).

Lemma 3.8

Consider the following map, which evaluates a vector

(x_{1}, x_{2}, \dots, x_{n})

at all elementary symmetric polynomials on the set

{x_{1}, x_{2}, \dots, x_{n}}

:

\begin{matrix} R^{n} & \to R^{n} \\ (x_{1}, x_{2}, \dots, x_{n}) & \mapsto (e_{1} ({x_{1}, x_{2}, \dots, x_{n}}), e_{2} ({x_{1}, x_{2}, \dots, x_{n}}),) \\ (\dots, e_{n} ({x_{1}, x_{2}, \dots, x_{n}})) . \end{matrix}

Then this map is generically n!-to-1.

Identifiability of $M_{n} (1, 1)$

In this subsection, we show that all parameters are SLING in the mammillary model with input and output in the central compartment (see Figure 8).

**Fig. 8** Parameter identifiability for $M_{n} (1, 1)$ : all parameters are SLING

Proposition 3.9

( $M_{n} (1, 1)$ ) Let $n \geq 3$ . If $M = (G, In, Out, Leak)$ is an n-compartment mammillary model with no leaks ( $Leak = \emptyset$ ) and input and output in the central compartment ( $In = Out = {1}$ ), then every parameter of $M$ is SLING.

Proof

Every permutation of ${2, 3, \dots, n}$ is a model automorphism of $M$ . Thus, the “incoming” edges $k_{12}, k_{13}, \dots, k_{1 n}$ are symmetrically identical, as are the “outgoing” edges $k_{21}, k_{31}, \dots, k_{n 1}$ . Hence (and here $n \geq 3$ is used), Lemma 2.32 implies that every parameter is SLING or unidentifiable. However, Remark 3.1 implies that all parameters are at least locally identifiable. We conclude that all parameters are SLING. $□$

Remark 3.10

Proposition 3.9 was proven previously (Cobelli et al. 1979) (using other methods), but we provide a proof here for completeness.

Identifiability of $M_{n} (1, 2)$

For the 4-compartment model

M_{4} (1, 2)

, from Figure 2, we saw in Example 2.26 that the parameter

k_{21}

is globally identifiable and all other parameters are SLING. This result generalizes to allow for more compartments, as follows (see Figure 9).

**Fig. 9** Parameter identifiability for $M_{n} (1, 2)$ : the (green) boldface arrow indicates a globally identifiable parameter, and all others are SLING

Proposition 3.11

(

M_{n} (1, 2)

) Let

n \geq 4

. If

M = (G, In, Out, Leak)

is an n-compartment mammillary model with

In = {1}

,

Out = {2}

, and

Leak = \emptyset

, then

the parameter $k_{21}$ is globally identifiable,
the parameter $k_{12}$ is SLING, and
the parameters $k_{13}, k_{14}, \dots, k_{1 n}, k_{31}, k_{41}, \dots, k_{n 1}$ are SLING.

Proof

Part (1) follows directly from Lemma 2.27. Also, part (3) follows from Remark 3.1, Lemma 2.32, and the fact that every permutation of ${3, 4, \dots, n}$ is a model automorphism of $M$ (here, the assumption $n \geq 4$ is used).

For part (2), we begin by analyzing the coefficients on the right-hand side of the input-output equation. By Proposition 2.11, these coefficients (denoted by

d_{i}

) are sums over certain subgraphs of the following graph

G_{2}^{*}

, which we obtain from the star graph in Figure 3 by removing the (unique) edge outgoing from compartment-2 (the output compartment):

Such subgraphs must connect the input and output compartments (in compartment-1 and compartment-2, respectively) and so must contain the edge

k_{21}

. Next, the condition of being an incoming forest precludes the remaining outgoing edges from compartment-1 (namely,

k_{31}, k_{41}, \dots, k_{n 1}

). Thus, the coefficients

d_{i}

are given by the following formulas involving elementary symmetric polynomials on the set

Σ : = {k_{13}, k_{14}, \dots, k_{1 n}}

:

\begin{matrix} d_{n - 2} & = k_{21} \\ d_{n - 3} & = k_{21} e_{1} (Σ) \\ d_{n - 4} & = k_{21} e_{2} (Σ) \\ ⋮ \\ d_{0} & = k_{21} e_{n - 2} (Σ) . \end{matrix}

21

Next, Proposition 3.7 gives the following equation, from (17), involving left-hand side coefficients

c_{i}

, the parameters

k_{12}

and

k_{21}

, and elementary symmetric polynomials on

Σ

:

\begin{matrix} k_{12}^{n - 2} c_{n - 1} - k_{12}^{n - 3} c_{n - 2} + k_{12}^{n - 4} c_{n - 3} - \dots \pm c_{1} \\ = (k_{12}^{n - 2} - k_{12}^{n - 3} e_{1} (Σ) + k_{12}^{n - 4} e_{2} (Σ) - \dots \pm e_{n - 2} (Σ)) k_{21} + k_{12}^{n - 1} . \end{matrix}

22

By moving all terms of equation (22) to the right-hand side, we obtain the following equation:

\begin{matrix} 0 & = k_{12}^{n - 1} + (k_{21} - c_{n - 1}) k_{12}^{n - 2} - (k_{21} e_{1} (Σ) - c_{n - 2}) k_{12}^{n - 3} + \dots \pm (k_{21} e_{n - 2} (Σ) - c_{1}) \\ = k_{12}^{n - 1} + \sum_{i = 2}^{n} {(- 1)}^{i} (k_{21} e_{i - 2} (Σ) - c_{n - i + 1}) k_{12}^{n - i}, \end{matrix}

23

where we use the fact that

e_{0} (Σ) : = 1

. Our aim is to apply Proposition 2.25 using the following function which is closely related to the right-hand side of (23):

\begin{matrix} h & : = z^{n - 1} + \sum_{i = 2}^{n} {(- 1)}^{i} (k_{21} e_{i - 2} (Σ) - c_{n - i + 1}) z^{n - i} . \end{matrix}

24

By construction (see equations (23)–(24)),

{h |}_{z = k_{12}} = 0

. Also, the leading coefficient (with respect to z) is 1, which is a nonzero function that is identifiable from the coefficient map (as in Definition 2.23). Therefore, to apply Proposition 2.25, we need only show that for all

i = 2, 3, \dots, n

, the polynomial function

{(- 1)}^{i} (k_{21} e_{i - 2} (Σ) - c_{n - i + 1})

is identifiable from the coefficient map. Indeed, the coefficients

c_{i}

are (by definition) identifiable from the coefficient map, and we see from (21) that

k_{21}

and the elementary symmetric polynomials

e_{1} (Σ), e_{2} (Σ), \dots, e_{n - 2} (Σ)

are identifiable from the coefficient map. Thus, we indeed can apply Proposition 2.25 and thereby conclude that

k_{12}

is generically locally identifiable.

It remains only to show that $k_{12}$ is not generically globally identifiable. By Definition 2.18, it suffices to find a positive-measure subset $Ω$ of the parameter space $R^{| E | + | Leak |} = R^{2 n}$ , such that, on $Ω$ , the relevant set arising from the coefficient map (namely, (9)) has size at least two.

As a step toward defining such a set

Ω

, we revisit the function h, from (24). We rewrite h by using the equations (15) in straightforward manner to obtain the first equality here:

\begin{matrix} h & = z^{n - 1} + \sum_{i = 2}^{n} {(- 1)}^{i + 1} (k_{12} g_{i - 2} + g_{i - 1}) z^{n - i} \\ = (z - k_{12}) (z^{n - 2} - g_{1} z^{n - 3} + g_{2} z^{n - 4} - \dots \pm g_{n - 2}), \end{matrix}

25

and the second equality above uses the fact that

g_{0} = g_{n - 1} = 0

(from Proposition 3.7).

Our next aim is to show that there is an open subset of parameter space $R^{2 n}$ on which, informally speaking, h always specializes to a (univariate) polynomial with $n - 1$ distinct real roots. More precisely, we will show the existence of an open set $Θ \subseteq R^{2 n}$ such that, for all $k^{*} \in Θ$ , the polynomial ${h |}_{k = k^{*}}$ (which is univariate in z) has $n - 1$ distinct real roots.

We will construct such a set

Θ

as an open neighborhood of some vector

k^{∙}

. To this end, we choose a vector of parameters

k^{∙} = (k_{12}^{∙}, k_{21}^{∙}, k_{13}^{∙}, k_{31}^{∙}, \dots, k_{1 n}^{∙}, k_{n 1}^{∙}) \in R^{2 n}

with two properties:

(i) $k_{31}^{∙} = k_{41}^{∙} = \dots = k_{n 1}^{∙} = 0$ (that is, all “outgoing” parameters, except $k_{21}$ , are 0), and
(ii) the “incoming” parameters $k_{12}^{∙}, k_{13}^{∙}, k_{14}^{∙}, \dots, k_{1 n}^{∙}$ are distinct.

We claim that

g_{i} |_{k = k^{∙}} = e_{i} (Σ^{∙})

for all

i = 1, 2, \dots, n - 2

, where

Σ^{∙} : = {k_{13}^{∙}, k_{14}^{∙}, \dots, k_{1 n}^{∙}}

. We verify this claim, as follows. First, recall from (16) that

g_{i}

is a sum over incoming forests of the graph

\tilde{G}

in Figure 7. Thus, evaluating at

k^{∙}

essentially disregards all “outgoing” edges in

\tilde{G}

(because the corresponding parameters are set to 0 by property (i)), leaving only “incoming” edges. These “incoming” edges correspond to the labels in

Σ^{∙}

, which can be chosen freely to form incoming forests. We conclude that

g_{i} |_{k = k^{∙}} = e_{i} (Σ^{∙})

, as claimed.

Using (25), it now follows that:

\begin{matrix} {h |}_{k = k^{∙}} = (z - k_{12}^{∙}) (z^{n - 2} - e_{1} (Σ^{∙}) z^{n - 3} + e_{2} (Σ^{∙}) z^{n - 4} - \dots \pm e_{n - 2} (Σ^{∙})) . \end{matrix}

26

It is easily seen from (26) that the set of roots of the polynomial

{h |}_{k = k^{∙}}

is

{k_{12}^{∙}} \cup Σ^{∙}

and therefore

{h |}_{k = k^{∙}}

has

n - 1

distinct real roots (property (ii) of

k^{∙}

is used here). We claim that there is an open neighborhood

Θ

of

k^{∙}

such that for every

k^{*} \in Θ

, the polynomial

{h |}_{k = k^{*}}

has

n - 1

distinct real roots. Indeed, this follows readily from two well-known facts: (1) the roots of a polynomial vary continuously in the coefficients of the polynomial (see, e.g., Rahman and Schmeisser 2002, Theorem 1.3.1, page 10), and (2) non-real roots of real-coefficient univariate polynomials come in complex-conjugate pairs.

Next, if needed, we shrink $Θ$ to a smaller open neighborhood $Ω$ of $k^{∙}$ so that property (ii) holds in the neighborhood. More precisely, we ensure that, for all $k^{*}$ in the neighborhood $Ω$ , the parameter values $k_{12}^{*}, k_{13}^{*}, \dots, k_{1 n}^{*}$ are distinct.

Now consider an arbitrary parameter vector

k^{*}

in

Ω

. By Definition 2.18, it suffices to show the existence of another parameter vector

k^{* *} \in R^{2 n}

, with

k_{12}^{* *} \neq k_{12}^{*}

, such that both parameter vectors have the same image under the coefficient map, that is,

\begin{matrix} c_{i} (k^{*}) = c_{i} (k^{* *}) and d_{j} (k^{*}) = d_{j} (k^{* *}) \end{matrix}

27

for all

1 \leq i \leq n - 1

and

0 \leq j \leq n - 2

, where the

c_{i}

’s and

d_{j}

’s are as in (15) and (21), respectively.

To this end, we begin to define such a vector

k^{* *}

by setting some of its coordinates equal to the corresponding values in

k^{*}

, as follows:

\begin{matrix} k_{21}^{* *} : = k_{21}^{*} and (k_{13}^{* *}, k_{14}^{* *}, \dots, k_{1 n}^{* *}) : = (k_{13}^{*}, k_{14}^{*}, \dots, k_{1 n}^{*}) . \end{matrix}

28

The above choice for

k_{21}^{* *}

is mandatory, as

k_{21}

is globally identifiable. Also, the choices (28) guarantee that the parameters appearing in the formulas for

d_{0}, d_{1}, \dots, d_{n - 2}

, in (21), are the same in

k^{*}

and

k^{* *}

. Therefore, the desired equalities

d_{j} (k^{*}) = d_{j} (k^{* *})

, from (27), hold for all

0 \leq j \leq n - 2

.

The rest of this proof consists of defining the remaining coordinates of $k^{* *}$ and then verifying the desired equalities $c_{i} (k^{*}) = c_{i} (k^{* *})$ from (27).

To this end, consider the univariate polynomial

{h |}_{k = k^{*}}

. One of its roots, by (25), is

k_{12}^{*}

. Additionally, there are

n - 2

additional (distinct) real roots (by construction of

Ω

). Accordingly, we pick one of these roots and set

k_{12}^{* *}

equal to it. Hence, we have:

\begin{matrix} {h |}_{k = k^{*}} (k_{12}^{* *}) = 0 and k_{12}^{* *} \neq k_{12}^{*} . \end{matrix}

29

To define the remaining entries of

k^{* *}

, we need to use a matrix that is obtained by specializing and performing row-operations on the matrix M in (19). To give details, consider the following

(n - 2) \times (n - 2)

matrix:

\begin{matrix} M^{*} {: = M |}_{k_{13} = k_{13}^{*}, k_{14} = k_{14}^{*}, \dots, k_{1 n} = k_{1 n}^{*}} . \end{matrix}

30

Next, let

Shift (M^{*})

be the

(n - 2) \times (n - 2)

matrix obtained from

M^{*}

by shifting the rows down by one; more precisely, the first row of

Shift (M^{*})

consists of 0’s, the second row is the first row of

M^{*}

, the third row is the second row of

M^{*}

, and so on, ending with row-

(n - 2)

being the row-

(n - 3)

of

M^{*}

. Finally, define:

\begin{matrix} \tilde{M} : = M^{*} + k_{12}^{*} Shift (M^{*}) . \end{matrix}

31

We claim that

\tilde{M}

is invertible. To see this, recall from Proposition 3.7 that

det M = \pm \prod_{3 \leq j < ℓ \leq n} (k_{1 j} - k_{1 ℓ})

. Therefore, by (30),

det M^{*} = \pm \prod_{3 \leq j < ℓ \leq n} (k_{1 j}^{*} - k_{1 ℓ}^{*})

, which is nonzero (recall that the fact that

k^{*}

is in

Ω

guarantees that

k_{13}^{*}, k_{14}^{*}, \dots, k_{1 n}^{*}

are distinct). Hence, the matrix

M^{*}

is invertible. Finally, by construction (31),

\tilde{M}

is obtained from

M^{*}

by row operations and so is also invertible.

As

\tilde{M}

is invertible, we can now define the remaining entries of

k^{* *}

, as follows, where we use the notation

Σ^{*} : = {k_{13}^{*}, k_{14}^{*}, \dots, k_{1 n}^{*}}

:

Image not available

32

Next, we verify the equalities

c_{i} (k^{*}) = c_{i} (k^{* *})

, in (27), for

2 \leq i \leq n - 1

. Indeed, it is straightforward (although tedious) to check that these equalities follow readily from the formulas for

c_{2}, c_{3}, \dots, c_{n - 1}

in (15), the formulas for the

g_{i}

’s in terms of the matrix M in (18), the construction of the matrix

\tilde{M}

in (31), and our choice of parameters

k_{ij}^{* *}

, including the formula (32).

Finally, it remains only to check that

c_{1} (k^{*}) = c_{1} (k^{* *})

. To see this, we first recall from (29) that

{h |}_{k = k^{*}} (k_{12}^{* *}) = 0

. Thus, by using (24) (and comparing with (23)), we obtain the following:

\begin{matrix} 0 & = {(k_{12}^{* *})}^{n - 1} + (k_{21}^{*} - c_{n - 1}^{*} (k^{*})) {(k_{12}^{* *})}^{n - 2} - (k_{21}^{*} e_{1} (Σ^{*}) - c_{n - 2}^{*} (k^{*})) {(k_{12}^{* *})}^{n - 3} + \dots \\ \pm (k_{21}^{*} e_{n - 2} (Σ^{*}) - c_{1}^{*} (k^{*})) . \end{matrix}

33

Next, consider specializing the equation (23) at

k = k^{* *}

. We claim that the resulting equation is exactly what is shown in (33), except that

c_{1} (k^{*})

(in the second line) is replaced by

c_{1} (k^{* *})

. Indeed, this claim follows readily from two facts: (1) the parameter

k_{21}

and the parameters in

Σ

are the same between

k^{*}

and

k^{* *}

(recall (28)), and (2)

c_{i} (k^{*}) = c_{i} (k^{* *})

, for

2 \leq i \leq n - 1

, which we showed above. We therefore conclude that

c_{1} (k^{*}) = c_{1} (k^{* *})

.

□

Identifiability of $M_{n} (2, 1)$

In this subsection, we analyze mammillary models with input in a peripheral compartment and output in the central compartment (Figure 10). The proof of Proposition 3.12 below is similar to certain parts of the proof we gave above for Proposition 3.11.

**Fig. 10** Parameter identifiability for $M_{n} (2, 1)$ : the (green) boldface arrows indicate generically globally identifiable parameters (in fact, $k_{12}$ is globally identifiable), and all others are SLING

Proposition 3.12

(

M_{n} (2, 1)

) Let

n \geq 4

. If

M = (G, In, Out, Leak)

is an n-compartment mammillary model with

In = {2}

,

Out = {1}

, and

Leak = \emptyset

, then

the parameter $k_{12}$ is globally identifiable,
the parameter $k_{21}$ is generically globally identifiable, and
the parameters $k_{13}, k_{14}, \dots, k_{1 n}, k_{31}, k_{41}, \dots, k_{n 1}$ are SLING.

Proof

Part (1) follows directly from Lemma 2.27 ( $k_{12}$ equals the coefficient $d_{n - 2}$ ). Also, part (3) follows from Remark 3.1, Lemma 2.32, and the fact that every permutation of ${3, 4, \dots, n}$ is a model automorphism of $M$ (here, the assumption $n \geq 4$ is used).

For part (2), we begin by analyzing the coefficients on the right-hand side of the input-output equation. According to Proposition 2.11, these coefficients are sums over certain subgraphs of the following graph

G_{1}^{*}

, which is obtained from the star graph in Figure 3 by removing all the edges outgoing from compartment-1 (the output compartment):

In order to connect the input and output compartments (in compartment-2 and compartment-1, respectively), the subgraphs must contain the edge

k_{12}

. The condition of being an incoming forest is vacuous (in the graph above, no compartment has more than one outgoing edge, nor can any of the edges form a cycle). We conclude that

k_{12}

is a factor of every right-hand side coefficient

d_{i}

, as in (8), and moreover these coefficients

d_{i}

are given by the following formulas involving elementary symmetric polynomials on the set

Σ : = {k_{13}, k_{14}, \dots, k_{1 n}}

:

\begin{matrix} d_{n - 2} & = k_{12} \\ d_{n - 3} & = k_{12} e_{1} (Σ) \\ d_{n - 4} & = k_{12} e_{2} (Σ) \\ ⋮ \\ d_{0} & = k_{12} e_{n - 2} (Σ) = k_{12} k_{13} \dots k_{1 n} . \end{matrix}

34

Next, Proposition 3.7 yields an equation (17) involving left-hand side coefficients

c_{i}

, which we reproduce here for convenience:

\begin{matrix} k_{12}^{n - 2} c_{n - 1} - k_{12}^{n - 3} c_{n - 2} + k_{12}^{n - 4} c_{n - 3} - \dots \pm c_{1} \\ = (k_{12}^{n - 2} - k_{12}^{n - 3} e_{1} (Σ) + k_{12}^{n - 4} e_{2} (Σ) - \dots \pm e_{n - 2} (Σ)) k_{21} + k_{12}^{n - 1} . \end{matrix}

35

The idea now is to view equation (35) as a linear polynomial in

k_{21}

with coefficients that are identifiable from the coefficient map. More precisely, we use the equations (34) to perform the following steps to obtain a nonzero linear polynomial in

k_{21}

whose coefficients are polynomial expressions in the

c_{i}

’s and

d_{i}

’s (and thus are identifiable from the coefficient map):

Multiply both sides by $k_{12}$ , and then replace $k_{12}^{n - 2} e_{1} (Σ)$ by $k_{12}^{n - 3} d_{n - 3}$ , and so on, ending by replacing $k_{12} e_{n - 2} (Σ)$ by $d_{0}$ ;
Replace each occurrence of $k_{12}$ by $d_{n - 2}$ .

Now Proposition 2.25 applies, and so

k_{21}

is generically globally identifiable.

□

Identifiability of $M_{n} (2, 2)$

This subsection focuses on mammillary models in which the input and output are in the same peripheral compartment. We show that the edges incident to this peripheral compartment have generically globally identifiable parameters, and all other parameters are SLING (see Figure 11). We motivate this result – and the idea behind the proof – through the following example.

**Fig. 11** Parameter identifiability for $M_{n} (2, 2)$ : the (green) boldface arrows indicate generically globally identifiable parameters (in fact, $k_{12}$ is globally identifiable), and all others are SLING

Example 3.13

By Proposition 2.11, the coefficients of the input-output equation for the model

M_{4} (2, 2)

are as follows:

\begin{matrix} c_{3} & = k_{12} + k_{13} + k_{14} + k_{21} + k_{31} + k_{41} \\ c_{2} & = k_{12} k_{13} + k_{12} k_{14} + k_{12} k_{31} + k_{12} k_{41} + k_{13} k_{14} + k_{13} k_{21} \\ + k_{13} k_{41} + k_{14} k_{21} + k_{14} k_{31} \\ c_{1} & = k_{12} k_{13} k_{14} + k_{12} k_{13} k_{41} + k_{12} k_{14} k_{31} + k_{13} k_{14} k_{21} \\ d_{2} & = k_{13} + k_{14} + k_{21} + k_{31} + k_{41} \\ d_{1} & = k_{13} k_{14} + k_{13} k_{21} + k_{13} k_{41} + k_{14} k_{21} + k_{14} k_{31} \\ d_{0} & = k_{13} k_{14} k_{21} \end{matrix}

It is straightforward to check that the difference between the two linear coefficients,

c_{3} - d_{2}

, equals

k_{12}

and so this parameter is globally identifiable. Additionally, the difference between the degree-two coefficients,

c_{2} - d_{1}

, is a multiple of

k_{12}

and, indeed, equals

k_{12} (d_{2} - k_{21})

. As we already saw that

k_{12}

is globally identifiable, we conclude that

k_{21}

is generically globally identifiable (“generically” is here because, when

k_{12} = 0

, it may not be possible to recover

k_{21}

). These ideas generalize, as seen in the next result (equation (36) in particular).

Proposition 3.14

(

M_{n} (2, 2)

) Let

n \geq 4

. If

M = (G, In, Out, Leak)

is an n-compartment mammillary model with

In = {2}

,

Out = {2}

, and

Leak = \emptyset

, then

the parameters $k_{12}$ and $k_{21}$ are generically globally identifiable (in fact, $k_{12}$ is globally identifiable), and, moreover, their values can be determined by the following formulas:
$\begin{matrix} k_{12} & = c_{n - 1} - d_{n - 2}, and \\ k_{21} & = d_{n - 2} - \frac{c_{n - 2} - d_{n - 3}}{k_{12}} = d_{n - 2} - \frac{c_{n - 2} - d_{n - 3}}{c_{n - 1} - d_{n - 2}}, \end{matrix}$ 36
where $c_{i}$ and $d_{i}$ are the coefficients of the input-output equation, as in (7)–(8); and
the parameters $k_{13}, k_{14}, \dots, k_{1 n}, k_{31}, k_{41}, \dots, k_{n 1}$ are SLING.

Proof

Part (2) follows from Remark 3.1, Lemma 2.32, and the fact that every permutation of ${3, 4, \dots, n}$ is a model automorphism of $M$ (here, the assumption $n \geq 4$ is used).

For part (1), we will use the formula for coefficents of the input-output equation (Proposition 2.11). In our model $M = (G, In, Out, Leak)$ , the input and output are in the same compartment, so it is vacuously true that the input and output are always connected in spanning subgraphs of $G_{2}^{*}$ (recall that $G_{2}^{*}$ is obtained from G by removing the outgoing edge $k_{12}$ from the output compartment-2). Therefore, the subgraph-connectedness condition for the right-hand side coefficients $d_{i}$ , in (8), can be safely ignored in what follows.

We first analyze $k_{12}$ . By Proposition 2.11, the linear-polynomial coefficient on the left-hand side, namely, $c_{n - 1}$ , is a sum over all edges of G. On the other hand, the corresponding coefficient on the right-hand side, $d_{n - 2}$ , is a sum over all edges except $k_{12}$ (because it was removed from G to obtain $G_{2}^{*}$ ). We conclude that $k_{12} = c_{n - 1} - d_{n - 2}$ , as desired.

Now we consider

k_{21}

. This time we focus on the degree-two coefficients in Proposition 2.11, namely,

c_{n - 2}

and

d_{n - 3}

. Their difference is a sum over all 2-edge spanning incoming forests of the mammillary graph such that one of the edges is

k_{12}

. The remaining edge in such a subgraph is any edge besides

k_{12}

and

k_{21}

. We therefore obtain the first equality here:

\begin{matrix} c_{n - 2} - d_{n - 3} & = k_{12} (k_{13} + k_{14} + \dots + k_{1 n} + k_{31} + k_{41} + \dots + k_{n 1}) \\ = k_{12} (d_{n - 2} - k_{21}), \end{matrix}

37

and the second equality above follows from the description of

k_{21}

given earlier in the proof.

Now it is straightforward to solve for $k_{21}$ in (37) and obtain the desired formula (36). $□$

Identifiability of $M_{n} (2, 3)$

As noted earlier in Remark 3.1, the model $M_{n} (2, 3)$ contains unidentifiable parameters. Some parameters, however, are SLING; these parameters correspond to certain edges directed toward to the central compartment (see the [blue] solid arrows in Figure 12). To illustrate the ideas behind the proof of this result (Proposition 3.16 below), we present the following example.

Example 3.15

For the model

M_{5} (2, 3)

, it is straightforward to apply Proposition 2.11 to obtain the following coefficients on the right-hand side of the input-output equation:

\begin{matrix} d_{3} & = 0 \\ d_{2} & = k_{12} k_{31} \\ d_{1} & = k_{12} k_{31} (k_{14} + k_{15}) \\ d_{0} & = k_{12} k_{31} (k_{14} k_{15}) . \end{matrix}

38

We see from the coefficients (38) that the following elementary symmetric polynomials are identifiable from the coefficient map:

e_{1} (Σ^{'}) = k_{14} + k_{15}

and

e_{2} (Σ^{'}) = k_{14} k_{15}

, where

Σ^{'} = {k_{14}, k_{15}}

. This allows us to recover the set

Σ^{'}

, but the symmetry between compartment-4 and compartment-5 precludes us from distinguishing between

k_{14}

and

k_{15}

. We conclude that the parameters

k_{14}

and

k_{15}

are SLING.

Proposition 3.16

( $M_{n} (2, 3)$ ) Let $n \geq 5$ . If $M = (G, In, Out, Leak)$ is an n-compartment mammillary model with $In = {2}$ , $Out = {3}$ , and $Leak = \emptyset$ , then the parameters $k_{14}, k_{15}, \dots, k_{1 n}$ are SLING.

Proof

Every permutation of ${4, 5, \dots, n}$ is a model automorphism of $M$ . Hence, by Proposition 2.32, the parameters $k_{14}, k_{15}, \dots, k_{1 n}$ are either all SLING or all unidentifiable (here, the assumption $n \geq 5$ is used). Therefore, it suffices to show that $k_{14}, k_{15}, \dots, k_{1 n}$ are generically locally identifiable.

Let

Σ^{'} : = {k_{14}, k_{15}, \dots, k_{1 n}}

. We claim that the right-hand side coefficients,

d_{i}

, as in (8), can be written in terms of elementary symmetric polynomials on

Σ^{'}

, as follows:

\begin{matrix} d_{n - 2} & = 0 \\ d_{n - 3} & = k_{12} k_{31} \\ d_{n - 4} & = k_{12} k_{31} e_{1} (Σ^{'}) \\ d_{n - 5} & = k_{12} k_{31} e_{2} (Σ^{'}) \\ ⋮ \\ d_{0} & = k_{12} k_{31} e_{n - 3} (Σ^{'}) . \end{matrix}

39

(These equations (39) in the case of

n = 5

were shown earlier in (38).)

We verify the formulas (39) by using Proposition 2.11, as follows. First, the input and output are in distinct peripheral compartments, so they can not be connected by a single edge. Thus, $d_{n - 2} = 0$ . Next, the only spanning incoming forests with two edges that connect the input to the output here are $k_{12} k_{31}$ and $k_{21} k_{13}$ , and since the edge $k_{13}$ is deleted in $G^{*}$ , we must have $d_{n - 3} = k_{12} k_{31}$ . Similarly, all terms in the remaining coefficients $d_{i}$ , for $i = n - 2, n - 1, \dots, 0$ , must have a factor of $k_{12} k_{31}$ , since this is the only pair of edges that can take part in a spanning incoming forest that connects the input and output. If we begin with $k_{12}$ and $k_{31}$ and then try to add more edges, the only edges we can add without creating cycles or having more than one outgoing edge from compartment-1 are the edges of $Σ^{'} = {k_{14}, k_{15}, \dots, k_{1 n}}$ . Indeed, such edges can be added arbitrarily, which yields the elementary symmetric polynomials shown in (39).

Next, the equations (39) imply that the elementary symmetric polynomials appearing in these equations, namely, $e_{1} (Σ^{'}), e_{2} (Σ^{'}), \dots, e_{n - 3} (Σ^{'})$ , are identifiable from the coefficient map. Now Lemma 3.8 implies that each parameter $k_{14}, k_{15}, \dots, k_{1 n}$ , when viewed as a (projection) function $R^{| E | + | Leak |} \to R$ , is locally identifiable from the coefficient map. Hence, by Remark 2.24, each of the parameters $k_{14}, k_{15}, \dots, k_{1 n}$ is generically locally identifiable. $□$

Remark 3.17

( $M_{4} (2, 3)$ ) The ideas in the proof of Proposition 3.16 can be used to show that, in the $n = 4$ version of the model $M_{n} (2, 3)$ , the parameter $k_{14}$ is generically globally identifiable. Indeed, in this case, the equations (39) imply that $k_{14} = \frac{d_{0}}{d_{1}}$ .

Next, we conjecture that all parameters not considered in Proposition 3.16 are unidentifiable, as follows (cf. Figure 12).

**Fig. 12** Parameter identifiability for $M_{n} (2, 3)$ : the red dashed arrows indicate parameters that we conjecture to be unidentifiable, and all others are SLING

Conjecture 3.18

(

M_{n} (2, 3)

) Let

n \geq 5

. If

M = (G, In, Out, Leak)

is an n-compartment mammillary model with

In = {2}

,

Out = {3}

, and

Leak = \emptyset

, then the following parameters are unidentifiable:

\begin{matrix} k_{12}, k_{21}, k_{13}, and k_{31}, k_{41}, \dots, k_{n 1} . \end{matrix}

Remark 3.19

By Lemma 2.32, the parameters $k_{41}, k_{51}, \dots, k_{n 1}$ in the model $M_{n} (2, 3)$ all have the same type of identifiability. Also, the equation $d_{n - 3} = k_{12} k_{31}$ , from equation (39) in the proof of Proposition 3.16, implies that $k_{12}$ and $k_{31}$ have the same type of identifiability. Conjecture 3.18 asserts that all of these parameters are, in fact, unidentifiable.

Discussion

In this work, we prove the identifiability properties of individual parameters (specifically, generically globally identifiable versus SLING) in mammillary models with one input, one output, and no leaks (Theorem 3.2). As noted earlier, we are essentially the first to undertake such an investigation for an infinite family of models. We therefore expect that our work to lead the way for future studies, involving other families of models.

A natural candidate for such future investigations is the family of catenary models, those in which the underlying graph is a bidirected path (see (Cobelli et al. 1979) and (Cobelli et al. 2002, §5.8) for partial results in this direction). A starting point for such a project is the database of catenary models with three compartments that appears in the recent work of Dessauer et al. (2024, Appendix B). In this database, each model is shown together with the identifiability properties of all parameters. We also direct the reader to additional databases of linear compartmental models, one for models with three compartments (created by Norton) (Norton 1982) and one for models with up to four compartments (created by Gogishvili) (Gogishvili 2024).

Returning our attention to mammillary models, there are a number of future directions. The first is to resolve Conjecture 3.18, which pertains to the parameters in model $M_{n} (2, 3)$ that we believe are unidentifiable (as shown in Figure 1). Additionally, it is natural to ask how our results extend to allow for (mammillary models with) any number of inputs, outputs, and leaks. In other words, how does Theorem 3.2 generalize when inputs, output, and/or leaks are added? This question is open, although some partial results were proven by Chau (1985b) and Cobelli et al. (1979) (see also (Cobelli et al. 2002, §5.8)).

A version of this question – at the level of models rather than their parameters – has been addressed in general (Chan et al. 2021; Gerberding et al. 2020; Gogishvili 2024; Gross et al. 2019), but we are not aware of results in this direction pertaining to individual parameters. The aforementioned questions – which investigate the effect on identifiability of various operations on models – are important in the context of systems biology, because a model might not be completely known or we may want to predict the effect of various interventions that lead to changes in the model. For instance, we might not know precisely which compartments have leaks, and so we would want to answer the question of identifiability for a family of models arising from various possible sets of leaks. Additionally, although we considered models without leaks in this work, we recognize that models that do contain leaks are more realistic in biological settings, as there are very few systems that are completely closed. Finally, allowing for more inputs or outputs represents additional controls or measurements of the system, which again may better reflect the situation in experimental settings. In short, our results take the first step into an investigation with direct relevance for biological applications.

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Acknowledgements

This research was initiated by Katherine Clemens, Jonathan Martinez, and Michaela Thompson at the 2023 REU in the Department of Mathematics at Texas A&M University, in which Anne Shiu and Benjamin Warren were research mentors. The REU was supported by the NSF (DMS-2150094); and AS and BW were partially supported by the NSF (DMS-1752672). The authors thank Cashous Bortner and Alejandro F. Villaverde for helpful feedback on an earlier version of this work. The authors also acknowledge an anonymous reviewer whose insightful comments improved this work.

Data Availability

This work does not have any associated data.

Declarations

Competing interests

The authors have no competing interests to declare that are relevant to the content of this article.

References

Ahmed S, Crepeau N, Dessauer Jr. PR, Edozie A, Garcia-Lopez O, Grimsley T, Lopez Garcia J, Neri V, Shiu A (2024) Identifiability of catenary and directed-cycle linear compartmental models. Preprint, arXiv:2412.05283
BarreiroXRVillaverdeAFOn the origins and rarity of locally but not globally identifiable parameters in biological modelingIEEE Access202311654576546710.1109/ACCESS.2023.3288998
Bortner C, Gilliana J, Patel D, Tamras Z (2024) Graph-based proofs of indistinguishability of linear compartmental models. Preprint, arXiv:2412.01135
BortnerCGrossEMeshkatNShiuASullivantSIdentifiability of linear compartmental tree models and a general formula for input-output equationsAdv Appl Math202314610249010.1016/j.aam.2023.102490
BortnerCMeshkatNIdentifiable paths and cycles in linear compartmental modelsB Math Biol20228455310.1007/s11538-022-01007-5PMC893402935305179
BortnerCMeshkatNGraph-based sufficient conditions for indistinguishability of linear compartmental modelsSIAM J Appl Dyn Syst20242332179220710.1137/23M1614663
Chan P, Johnston K, Shiu A, Sobieska A, Spinner C (2021) Identifiability of linear compartmental models: The impact of removing leaks and edges. Preprint, arXiv:2102.04417
ChauNPLinear n-compartment catenary models: formulas to describe tracer amount in any compartment and identification of parameters from a concentration-time curveMath Biosci198576218520610.1016/0025-5564(85)90004-5
ChauNPParameter identification in $N$ -compartment mamillary modelsMath Biosci198574219921810.1016/0025-5564(85)90055-0
CherruaultYSarinVBA six-compartment linear mammillary modelKybernetes199221232910.1108/eb005914
CobelliCLepschyAJacurGRIdentifiability results on some constrained compartmental systemsMath Biosci197947317319510.1016/0025-5564(79)90036-1
CobelliCFosterDToffoloGTracer kinetics in biomedical research: from data to model2002BerlinSpringer
DongRGoodbrakeCHarringtonHAPogudinGDifferential elimination for dynamical models via projections with applications to structural identifiabilitySIAM J Appl Algebra Geom20237119423510.1137/22M1469067
GerberdingSObatakeNShiuAIdentifiability of linear compartmental models: the effect of moving inputs, outputs, and leaksLinear Multilinear A202070142782280310.1080/03081087.2020.1812497
Godfrey K (1983) Compartmental models and their application. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London. With a foreword by A. G. J. MacFarlane
Gogishvili N (2024) Database for identifiability properties of linear compartmental models. Preprint, arXiv:2406.16132
GrossEHarringtonHMeshkatNShiuALinear compartmental models: input-output equations and operations that preserve identifiabilitySIAM J Appl Math20197941423144710.1137/18M1204826
GrossEMeshkatNShiuAIdentifiability of linear compartmental models: the singular locusAdv Appl Math2022133C10226810.1016/j.aam.2021.102268
HongHOvchinnikovAPogudinGYapCGlobal identifiability of differential modelsComm Pure Appl Math20187391831187910.1002/cpa.21921
HongHOvchinnikovAPogudinGYapCSIAN: software for structural identifiability analysis of ODE modelsBioinformatics201935162873287410.1093/bioinformatics/bty106930601937
Jacquez JA (1972) Compartmental analysis in biology and medicine. Kinetics of distribution of tacer-labeled materials. Amsterdam, New York, Elsevier Pub. Co
MassonisGBangaJRVillaverdeAFStructural identifiability and observability of compartmental models of the COVID-19 pandemicAnnu Rev Control20215144145910.1016/j.arcontrol.2020.12.00133362427PMC7752088
MatisJHKodellRLCardenasMA note on the use of a stochastic mammillary compartmental model as an environmental safety modelB Math Biol19763846747810.1016/S0092-8240(76)90001-X963330
Meshkat N, Rosen Z, Sullivant S (2018) Algebraic tools for the analysis of state space models. In The 50th anniversary of Gröbner bases, vol 77 of Adv. Stud. Pure Math., pp 171–205. Math. Soc. Japan, Tokyo
MeshkatNSullivantSEisenbergMIdentifiability results for several classes of linear compartment modelsB Math Biol20157781620165110.1007/s11538-015-0098-026337290
MulhollandRobert JKeenerMarvin SAnalysis of linear compartment models for ecosystemsJ Theor Biol197444110511610.1016/S0022-5193(74)80031-74822909
Muñoz-TamayoRPuilletLDanielJBSauvantDMartinOTaghipoorMBlavyPReview: To be or not to be an identifiable modelIs this a relevant question in animal science modelling? animal201812470171210.1017/S175173111700277429096725
NakashimaEBenetLZAn integrated approach to pharmacokinetic analysis for linear mammillary systems in which input and exit may occur in/from any compartmentJ Pharmacokinet Biop19891767368610.1007/BF010621242635739
NortonJPAn investigation of the sources of nonuniqueness in deterministic identifiabilityMath Biosci19826018910810.1016/0025-5564(82)90033-5
OvchinnikovAPillayAPogudinGScanlonTMulti-experiment parameter identifiability of ODEs and model theorySIAM J Appl Algebra Geom20226333936710.1137/21M1389845
OvchinnikovAPogudinGThompsonPInput-output equations and identifiability of linear ODE modelsIEEE T Automat Contr202368281282410.1109/TAC.2022.3145571
RahmanQISchmeisserGAnalytic theory of polynomials2002OxfordThe Clarendon Press, Oxford University Press
StigterJDComputing parameter identifiability and other structural properties for natural resource modelsNat Resour Model2024371e1238210.1111/nrm.12382
TangLZhouYWangLPurkayasthaSZhangLHeJWangFSongPX-KA review of multi-compartment infectious disease modelsInt Stat Rev202088246251310.1111/insr.1240232834402PMC7436714
WielandF-GHauberALRosenblattMTönsingCTimmerJOn structural and practical identifiabilityCurr Opin Syst Biol202125606910.1016/j.coisb.2021.03.005

Author Contributions

Pieter Verhoeven: Project administration, Conceptualization, Validation
Kenji Takahashi: Formal analysis, Funding acquisition

Declarations

Competing Interests

The authors declare no competing interests.

Data Availability

The datasets generated and analyzed during the study are available from the corresponding author on reasonable request.

Demonstration content: the full text of this article is adapted from “Parameter identifiability of linear-compartmental mammillary models” by Clemens and colleagues (Bulletin of Mathematical Biology, 2025; https://doi.org/10.1007/s11538-025-01568-1), which is available under a Creative Commons license (https://creativecommons.org/licenses/by/4.0/).

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License: https://creativecommons.org/licenses/by-nc/4.0/

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Abstract

Introduction

Background

Linear compartmental models

Definition 2.1

Definition 2.2

Definition 2.3

Example 2.4

Example 2.5

Input-output equations

Notation 2.6

Proposition 2.7

Example 2.8

Example 2.9

Definition 2.10

Proposition 2.11

Remark 2.12

Remark 2.13

Example 2.14

Identifiability of models

Definition 2.15

Proposition 2.16

Example 2.17

Identifiability of parameters

Definition 2.18

Notation 2.19

Remark 2.20

Remark 2.21

Remark 2.22

Identifiable functions

Definition 2.23

Remark 2.24

Proposition 2.25

Proof

Example 2.26

Lemma 2.27

Proof

Remark 2.28

Symmetric edge parameters

Example 2.29

Definition 2.30

Remark 2.31

Lemma 2.32

Proof

Results

Remark 3.1

Theorem 3.2

Remark 3.3

Preliminaries

Definition 3.4

Example 3.5

Example 3.6

Proposition 3.7

Proof

Lemma 3.8

Identifiability of Mn(1,1)

Proposition 3.9

Proof

Remark 3.10

Identifiability of Mn(1,2)

Proposition 3.11

Proof

Identifiability of Mn(2,1)

Proposition 3.12

Proof

Identifiability of Mn(2,2)

Example 3.13

Proposition 3.14

Proof

Identifiability of Mn(2,3)

Example 3.15

Proposition 3.16

Proof

Remark 3.17

Conjecture 3.18

Remark 3.19

Discussion

Acknowledgements

Data Availability

Declarations

Identifiability of $M_{n} (1, 1)$

Identifiability of $M_{n} (1, 2)$

Identifiability of $M_{n} (2, 1)$

Identifiability of $M_{n} (2, 2)$

Identifiability of $M_{n} (2, 3)$