Identifiability of Linear Compartmental Models: The Impact of Removing Leaks and Edges
Patrick Chan, Katherine Johnston, Anne Shiu, Aleksandra Sobieska, Clare Spinner
aa r X i v : . [ m a t h . D S ] F e b IDENTIFIABILITY OF LINEAR COMPARTMENTAL MODELS:THE IMPACT OF REMOVING LEAKS AND EDGES
PATRICK CHAN , KATHERINE JOHNSTON , ANNE SHIU , ALEKSANDRA SOBIESKA ,AND CLARE SPINNER Abstract
A mathematical model is identifiable if its parameters can be recovered fromdata. Here, we focus on a particular class of model, linear compartmental models,which are used to represent the transfer of substances in a system. We analyze whathappens to identifiability when operations are performed on a model, specifically,adding or deleting a leak or an edge. We first consider the conjecture of Gross etal. that states that removing a leak from an identifiable model yields a model thatis again identifiable. We prove a special case of this conjecture, and also show thatthe conjecture is equivalent to asserting that leak terms do not divide the so-calledsingular-locus equation. As for edge terms that do divide this equation, we conjec-ture that removing any one of these edges makes the model become unidentifiable,and then prove a case of this somewhat surprising conjecture. Introduction
A model is generically structurally identifiable if it has the following desirable property:from generic values of the inputs and initial conditions, the model parameters can be recov-ered from exact measurements of both inputs and outputs [2]. In this paper, we considerthe problem of assessing structural identifiability for a particular class of models, called lin-ear compartmental models . Linear compartmental models are used in a variety of fields,including in biological applications, to represent the transfer of substances in a system (e.g.,[5, 11, 13]). In the context of pharmacokinetics, a better understanding of such models mightreveal better drug dosing regimes and more precise measurements of drug transfers withinthe body.We focus on the question of how identifiability is affected when operations – such as addingor removing inputs, outputs, leaks, or edges – are performed on a model [4, 9, 8, 15, 16].Gross et al. proved that under certain hypotheses, adding or removing a leak preservesidentifiability [8]. Subsequently, Gerberding et al. proved related results – pertaining toadding or removing leaks and edges, and moving the output – for several common families ofmodels, specifically, catenary, cycle, and mammillary models [6]. In this work, we considermore general models, and again investigate the effect of adding or removing a leak or edge. Loyola University Chicago Lafayette College Texas A&M University University of Wisconsin-Madison University of Portland
Date : February 6, 2021.
Our two guiding questions, which were posed in earlier work [8, 10], are as follows. First,does deleting a leak from an identifiable model yield a model that is also identifiable? Thesecond question pertains to the singular-locus equation , which defines where the Jacobianmatrix of a model’s coefficient map (which is used for analyzing identifiability) drops rank. Itis known that deleting an edge whose corresponding parameter does not divide the singular-locus equation yields a submodel that is again identifiable [10]. The question here is, for anedge that does divide the singular-locus equation, is the resulting submodel unidentifiable?An answer would help clarify what information the singular-locus equation provides regardingthe identifiability of submodels [10].Both of the aforementioned questions are conjectured to have affirmative answers; seeConjecture 2.15, which was originally posed in [8], and Conjecture 2.20 (see also the sum-mary in Table 1). We prove several results toward these conjectures. First, we show a specialcase of an equivalent conjecture on adding leaks, when unidentifiability arises from havingtoo many parameters (Theorem 3.2). We also show that the conjecture on deleting leaks(Conjecture 2.15) is equivalent to asserting that leak terms do not divide the singular-locusequation (Theorem 3.8). Next, we prove that the conjecture on dividing edges (Conjec-ture 2.20) holds for models that satisfy certain combinatorial conditions (Theorem 4.1).Finally, we investigate the types of edges that divide the singular-locus equation.Operation Conjectured effect ResultRemove leak Identifiable Theorem 3.2Remove dividing edge Unidentifiable Theorem 4.1
Table 1.
Summary of conjectures and results on how operations affect iden-tifiability. For a strongly connected linear compartmental model M with atleast one input, if f M is obtained from M by the specified operation, and M isidentifiable, then f M is conjectured to be as listed in the second column. Par-tial results toward these conjectures are listed in the last column. For relatedprior results, see [6, Table 1] and [10, Tables 1 and 2].Our investigations into identifiability harness the theory of input-output equations forlinear compartmental models [1, 7, 14]. Our proofs therefore use linear-algebraic and com-binatorial arguments. We also rely on results of Bortner and Meshkat [4] and Mehskat,Sullivant, and Eisenberg [12].The outline of this paper is as follows. In Section 2, we introduce linear compartmentalmodels and identifiability. We prove our results pertaining to leaks and dividing edges inSections 3 and 4, respectively. We conclude with a discussion in Section 5.2. Background
This section introduces linear compartmental models, identifiability, and the singular lo-cus. We closely follow the notation in [6, 8, 10].2.1.
Linear compartmental models. A linear compartmental model ( G, In, Out, Leak )consists of a directed graph G = ( V, E ) and subsets
In, Out, Leak ⊆ V of (respectively) inputs, outputs, and leaks . Each i ∈ V is a compartment. Input compartments receive aninput stream, u i ( t ); and output compartments have an associated output measure, y ( t ). DENTIFIABILITY OF LINEAR COMPARTMENTAL MODELS 3
Assumption 2.1.
Throughout this work, we assume that every linear compartmental modelhas at least one output, because models without outputs are not identifiable.In keeping with the literature, output compartments are indicated using this symbol:. Input compartments are labeled “in”, and leaks are indicated by outgoing edges. Forexample, the linear compartmental model in Figure 1 has In = { } , Out = { } , and Leak = { , } . 1 2 3 k k k k in k k Figure 1.
A linear compartmental model.An edge j → i ∈ E represents flow from compartment j to compartment i . We assign a flow parameter k ij to each such edge. Leak compartments also have flow parameters, k ℓ ,which are the rates of flow exiting the system from compartment ℓ .Figure 1 depicts a three-compartment catenary model. From a biological standpoint, thismodel could represent the injection and flow of a drug within the body. The input is the drug,and compartment 1 is the injection site. Compartments 2 and 3 represent other organs in thebody where the drug travels. Edges between compartments 1, 2, and 3 indicate the transferof the drug between organs, and the leaks, labeled by parameters k and k , represent thetransfer of drugs from the measurable system into immeasurable parts of the body, such asthe bloodstream. The output is where the concentration of the drug is measured.We now introduce some more definitions. Definition 2.2.
A directed graph is strongly connected if there exists a path from eachvertex to every other vertex. A linear compartmental model (
G, In, Out, Leak ) is stronglyconnected if G is strongly connected. Definition 2.3.
For a linear compartmental model (
G, In, Out, Leak ) with n compartments,the compartmental matrix is the n × n matrix A given by the following: A ij := − k i − P p :( i → p ) ∈ E k pi if i = j and i ∈ Leak − P p :( i → p ) ∈ E k pi if i = j and i / ∈ Leakk ij if j → i is an edge of G G, In, Out, Leak ) defines the following system of ODEswith inputs u i ( t ) and outputs y i ( t ), where x ( t ) = ( x ( t ) , x ( t ) , . . . , x n ( t )) is the vector ofconcentrations in the compartments at time t : x ′ ( t ) = Ax ( t ) + u ( t ) , (1) y i ( t ) = x i ( t ) for all i ∈ Out , where u i ( t ) ≡ i / ∈ In . P. CHAN, K. JOHNSTON, A. SHIU, A. SOBIESKA, AND C. SPINNER
Example 2.4.
For the model in Figure 1, the ODEs are given by x ′ x ′ x ′ = − k − k k k − k − k − k k k − k x x x + u , with output equation y = x .2.2. Input-Output Equations.
Input-output equations of a linear compartmental modelare equations that hold along all solutions of the ODEs (1), and involve only the parame-ters k ij , input variables u i , output variables y i , and their derivatives. A general form of suchequations was proven by Meshkat, Sullivant, and Eisenberg [12, Theorem 2] (see also [8,Remark 2.7]), as follows. Proposition 2.5.
Consider a linear compartmental model M that is strongly connected andhas at least one input. Let A denote the compartmental matrix, let ∂ be the differentialoperator d/dt , and let ( ∂I − A ) ij be the submatrix of ( ∂I − A ) obtained by removing row i and column j . Then, for j ∈ Out , an input-output equation of M involving y j is: det( ∂I − A ) y j = X i ∈ In ( − i + j det ( ∂I − A ) ij u i . (2)We view the input-output equations (2) as polynomials in the y j ’s and u i ’s and theirderivatives. The coefficients of the equations are therefore polynomials in the parameters( k ℓm for edges m → ℓ , and k p for leaks p ∈ Leak ). The coefficients of the input-outputequations (2) that are non-constant polynomials are called the nontrivial coefficients . Definition 2.6.
For a linear compartmental model M , the coefficient map c : R | E | + | Leak | → R m sends the vector of parameters ( k ij ) to the vector of all nontrivial coefficients of allinput-ouput equations (2). Here, m denotes the number of such coefficients.1 2 43in k k k k k k Figure 2.
A linear compartmental model with In = { } , Out = { } , and Leak = ∅ . The dividing edges, k and k , are shown in blue (see Section 2.4). Example 2.7.
For the model in Figure 2, the compartmental matrix is: A = − k − k k k − k − k k k − k k k − k . DENTIFIABILITY OF LINEAR COMPARTMENTAL MODELS 5
The input-output equation (2), namely, det( ∂I − A ) y = − det ( ∂I − A ) u , is as follows:det ∂ + k + k − k − k ∂ + k + k − k − k ∂ + k − k − k ∂ + k y = det − k ∂ + k − k − k ∂ + k u , which expands to: y (4)1 + ( k + k + k + k + k + k ) y (3)1 (3) + ( k k + k k + k k + k k + k k + k k + k k + k k + k k + k k + k k + k k ) y (2)1 + ( k k k + k k k + k k k + k k k + k k k + k k k + k k k + k k k ) y ′ = k u (2)2 + k ( k + k ) u ′ + k k k u . The nontrivial coefficients in the input-output equation (3) form the coefficient map c : R → R ; for instance, c = k + k + k + k + k + k .The following conjecture addresses the size (the value of m ) of the coefficient map c : R | E | + | Leak | → R m for models with only one input and one output. Conjecture 2.8 (Number of coefficients) . Consider a strongly connected linear compartmen-tal model ( G, In, Out, Leak ) with | In | = | Out | = 1 . Let n be the number of compartments,and let L be the length of the shortest (directed) path in G from the input compartment tothe output compartment. Then, in the input-output equation (2) , the number of nontrivialcoefficients on the left-hand and right-hand sides are as follows: on LHS = ( n if Leak = ∅ n-1 if Leak = ∅ on RHS = ( n-1 if In = Out n-L if In = Out
One case of Conjecture 2.8 was proved by Bortner and Meshkat [4, Theorem 2.20], andmore cases will be resolved in forthcoming work of Bortner et al. [3]. Models that satisfyConjecture 2.8 and also the containment In ∪ Out ⊆ Leak are described by Bortner andMeshkat as having the “expected number of coefficients” [4, Definition 2.21].In this work, we are interested in what happens to identifiability when leaks or edges areadded or removed. The following lemma analyzes the effect on the coefficients.
Lemma 2.9 (Coefficients and adding a leak or edge) . Let M = ( G, In, Out, Leak ) be astrongly connected linear compartmental model with In = { i } and Out = { j } . Let n denotethe number of compartments. Let f M denote the model obtained from M obtained by adding aleak or an edge, with corresponding parameter k uv . Let d , d , . . . , d n +1 denote the coefficientof (respectively) y (0) j , y (1) j , . . . , y ( n ) j , u (0) i , u (1) i , . . . , u ( n − i in the input-output equation (2) for M . Let e d , e d , . . . , e d n +1 denote the corresponding coefficients (respectively) for f M . Then,for ℓ = 0 , , . . . , n + 1 , e d ℓ = d ℓ + k uv · g ℓ , for some polynomial g ℓ (with integer coefficients) in the flow parameters of M . P. CHAN, K. JOHNSTON, A. SHIU, A. SOBIESKA, AND C. SPINNER
Proof.
Let A and e A denote the compartmental matrices of M and f M , respectively. Notethat A = ( e A ) | k uv =0 . Now the result is immediate from equation (2). (cid:3) Identifiability.
A linear compartmental model is structurally identifiable if each of theparameters k ij can be recovered from data [2]. (In contrast, a model is practically identifiable if the parameters can be recovered from data with noise.) Our focus here is on structuralidentifiability. More precisely, we are interested in generic local identifiability, which refers toidentifiability except possibly for a measure-zero set of parameter space and also allows forrecovery of the parameters up to a finite set. This concept, in the case of strongly connectedmodels (and others as well), is captured by the following definition in terms of input-outputequations [14, Corollary 3.2]. Definition 2.10.
Let M = ( G, In, Out, Leak ) be a strongly connected linear compartmentalmodel with at least one input. Let c : R | E | + | Leak | → R m be the coefficient map derived fromthe input-output equations (2). Then, M is:(1) generically locally identifiable if, outside a set of measure zero, every point in R | E | + | Leak | has an open neighborhood U for which the restriction c | U : U → R m is one-to-one;(2) unidentifiable if c is infinite-to-one.Next, we recall the following useful criterion for (generic local) identifiability [12]. Proposition 2.11 (Meshkat, Sullivant, and Eisenberg [12]) . A linear compartmental model ( G, In, Out, Leak ) , with G = ( V, E ) , is generically locally identifiable if and only if the rankof the Jacobian matrix of its coefficient map, when evaluated at a generic point, is equal to | E | + | Leak | . Example 2.12 (Example 2.7, continued) . Continuing with the model in Figure 2, the Ja-cobian matrix of the coefficient map (arising from the input-output equation (3)) is: k + k + k k + k + k k + k + k + k + k k + k + k + k k + k + k + k + k k + k + k + k k k + k k k k + k k + k k k k + k k + k k + k k + k k k k + k k + k k + k k k k + k k + k k + k k + k k + k k k k + k k + k k + k k k + k k k k k k k k k The determinant of this matrix is the following nonzero polynomial (and so, for genericvalues of the k ij , the matrix has rank 5): k ( k + k − k − k )( k − k ) k . (4)It therefore follows from Proposition 2.11 that this model is generically locally identifiable.An important open problem is how to discern identifiability directly from a model. Thisquestion is challenging and subtle, as the models in Figures 3 and 4 show. The two modelsare very similar – the second model is obtained from the first by replacing the edge k with k – and yet the first model is identifiable while the second is unidentifiable. DENTIFIABILITY OF LINEAR COMPARTMENTAL MODELS 7 k k k k k k k k in Figure 3.
Model with In = Out = { } . The edges k , k ,and k , in blue, are dividingedges (see Section 2.4). 5 1 24 3 k k k k k k k k in Figure 4.
Model with In = Out = { } One way to be unidentifiable is to have too many edges or leaks (and thus too manyparameters). The following result in this direction is a special case of a result of Bortner andMeshkat [4, Theorem 6.1].
Proposition 2.13 (Unidentifiable from too many leaks) . Let M = ( G, In, Out, Leak ) be astrongly connected linear compartmental model with | In | = | Out | = 1 . If | Leak | > | In ∪ Out | ,then M is unidentifiable. One guiding question of our work, which was posed in [8, Question 5.2], is as follows.
Question 2.14.
Let f M be a linear compartmental model that is generically locally iden-tifiable and has at least one leak. If one leak is removed, is the resulting model M alwaysidentifiable?An affirmative answer to Question 2.14, under certain hypotheses, was conjectured byGross, Harrington, Meshkat, and Shiu, as follows [8, Conjecture 4.5]. Conjecture 2.15 (Remove leak) . Let f M be a strongly connected linear compartmental modelthat has at least one input and exactly one leak. If f M is generically locally identifiable, thenso is the model M obtained by removing the leak. Conjecture 2.15 holds in the following cases: • when f M has an input, output, and leak in a single compartment (and has no otherinputs, outputs, or leaks) [8, Proposition 4.6], and • when f M is obtained from an “identifiable cycle model” [12] by removing all leaksexcept one [6, Proposition 3.4].The first case above is, in fact, a special case of the second [4, Corollary 4.2]. Also, thissecond case includes some models that are widely used in applications, including certaincatenary, cycle, and mammillary models [10, Proposition 2.11]. P. CHAN, K. JOHNSTON, A. SHIU, A. SOBIESKA, AND C. SPINNER
Remark 2.16.
The converse to Conjecture 2.15 is true [8, Theorem 4.3].In the next section, we pose the contrapositive of Question 2.14, conjecture an affirmativeanswer, and then prove one case of this conjecture (Theorem 3.2). In doing so, we resolvea case of (the contrapositive of) Conjecture 2.15. We also prove that Conjecture 2.15 isequivalent to a new conjecture, which states that leak terms do not divide the singular-locusequation (see Theorem 3.8). We turn now to this topic of the singular locus.2.4.
Singular Locus.
Here we recall the definition of singular locus introduced in [10].
Definition 2.17.
Let M = ( G, In, Out, Leak ) be a strongly connected linear compartmentalmodel that has at least one input and is generically locally identifiable. Let c be the coefficientmap derived from the input-output equations (2). The singular locus is the subset of theparameter space R | E | + | Leak | where the Jacobian matrix of c has rank strictly less than | E | + | Leak | .Therefore, for identifiable linear compartmental models, the singular locus is defined bythe set of all ( | E | + | Leak | ) × ( | E | + | Leak | ) minors of the Jacobian matrix of the coefficientmap. For some models, this Jacobian matrix is square and hence there is only a single minorof interest, namely, the determinant. In this case, the determinant of the Jacobian matrix iscalled the singular-locus equation . (This equation is also defined in [10] for some non-squarecases, but here we mainly focus on the square case.)We call an edge j → i ∈ E a dividing edge if its parameter k ij divides the singular-locusequation; we also call k ij itself a dividing edge. We do not know whether the dividing edgescan be read directly from the model; this is an interesting future research direction. Example 2.18 (Example 2.12, continued) . For the model in Figure 2, the singular-locusequation was shown in equation (4). From this equation, we see that the dividing edges are k and k .Gross, Meshkat, and Shiu proved that removing non-dividing edges preserves identifiabil-ity, as follows [10, Theorem 3.1]. Proposition 2.19 (Deleting non-dividing edges [10]) . Let M = ( G, In, Out, Leak ) be astrongly connected linear compartmental model that is generically locally identifiable, withsingular-locus equation f . Let f M be the model obtained from M by deleting a set of edges I of G . If f M is strongly connected, and every edge in I is non-dividing, then f M is genericallylocally identifiable. Gross, Meshkat, and Shiu also asked whether a converse to Proposition 2.19 holds (see [10,Question 3.4]), and here we conjecture an affirmative answer, as follows.
Conjecture 2.20 (Remove dividing edge) . Let M be a strongly connected linear compart-mental model that is generically locally identifiable, with singular-locus equation f . If j → i is a dividing edge of M , and the model M ′ obtained by deleting the edge j → i is stronglyconnected, then M ′ is unidentifiable. Conjecture 2.20 is perhaps surprising; indeed, we initially expected a negative answerto [10, Question 3.4]. In a later section, we make progress toward Conjecture 2.20 (seeTheorem 4.1).
DENTIFIABILITY OF LINEAR COMPARTMENTAL MODELS 9
Remark 2.21 (Strong-connectedness hypothesis in Conjecture 2.20) . In Conjecture 2.20,the requirement that the model M ′ be strongly connected is so that (as mentioned earlier)the input-output approach to identifiability is valid [14]. Nevertheless, it is our observationthat in many cases the dividing edges of a model are such that the removal of such an edgewould break strong connectedness. This lack of strong connectedness means that, at leastintuitively, information is not fully flowing through the model from input to output, resultingin a situation that would lend itself to being unidentifiable. This is an interesting avenue forfuture research. 3. Results on leaks
In this section, we prove results on what happens to identifiability when leaks are added(Section 3.1) or removed (Section 3.2).3.1.
Adding a leak.
In this subsection, we address two questions concerning leaks. Thefirst, Question 2.14, was posed earlier (see Theorem 3.2). The second is the question of howmany added leaks force an identifiable model to become unidentifiable (see Theorem 3.4).Recall that Question 2.14 asked whether removing a leak from an identifiable model alwaysyields an identifiable model. That question is equivalent to asking whether adding a leakpreserves unidentifiability. We conjecture an affirmative answer, as follows.
Conjecture 3.1 (Add leak) . Let M be an unidentifiable linear compartmental model. Ifone leak is added, the resulting model f M is also unidentifiable. For strongly connected models, two subcases of Conjecture 3.1 naturally arise (throughProposition 2.11). In the first subcase, a model M is unidentifiable because all maximalminors of the Jacobian matrix are 0. In the second subcase, unidentifiability arises becausethe Jacobian matrix has more columns than rows (that is, the model has more parametersthan coefficients of the input-output equations), and so the rank of the m × ( | E | + | Leak | )Jacobian can not equal | E | + | Leak | . This second subcase is addressed in the following result. Theorem 3.2 (Add leak) . Let M be a strongly connected linear compartmental modelwith | In | = | Out | = 1 and coefficient map c : R | E | + | Leak | → R m . Consider the model f M formed by adding a single leak to M . Assume that Conjecture 2.8 holds for M and f M . If | E | + | Leak | > m and so M is unidentifiable, then f M is also unidentifiable.Proof. Assume that | E | + | Leak | (the number of parameters of M ) is strictly larger than m (the number of coefficients). For f M , the number of parameters is | E | + | Leak | + 1,while the number of coefficients is m (if Leak is nonempty) or m + 1 (if Leak is empty) byConjecture 2.8. Thus, by Proposition 2.11, f M is unidentifiable. (cid:3) Remark 3.3.
A recent result of Bortner and Meshkat is closely related to Theorem 3.2 andpertains to when | Leak | = | In ∪ Out | [4, Theorem 4.1].Next, we consider adding leaks to a model that may or may not be identifiable. Howmany leaks are too many, that is, how many leaks need to be added so that the model isautomatically unidentifiable? The following result addresses this question. Theorem 3.4 (Too many leaks) . Let M = ( G, In, Out, Leak ) be a strongly connected linearcompartmental model with at least one leak and | In | = | Out | = 1 . Let n be the number ofcompartments, and let L be the length of the shortest (directed) path in G from the input compartment to the output compartment. Assume that Conjecture 2.8 holds for M . If oneof the following holds: (I) In = Out and | Leak | > min { , n − | E | − } , or (II) In = Out and | Leak | > min { , n − | E | − L } ,then M is unidentifiable.Proof. If In = Out and | Leak | >
1, or if In = Out and | Leak | >
2, then Proposition 2.13implies that M is unidentifiable.Now assume that we are in the remaining cases, when In = Out and | Leak | > n −| E | − In = Out and | Leak | > n − | E | − L . The model M has | E | + | Leak | parameters.By Conjecture 2.8, the number of nontrivial coefficients on the left-hand side of the input-output equation (2) is n , while the number on the right-hand side is n − In = Out )or n − L (if In = Out ). It follows easily from our hypotheses on the number of leaks that | E | + | Leak | exceeds the number of coefficients, and so, by Proposition 2.11, the model M is unidentifiable. (cid:3) Example 3.5.
Let M = ( G, In, Out, Leak ) be a model for which G is the following cyclegraph: k k n,n − k k n k n Assume that In = { } and Out = { n } . It follows from Theorem 3.4 (assuming thatConjecture 2.8 holds for this model) that M is unidentifiable if there is more than oneleak (we have | E | = n and L = n − Removing a leak.
In this section we address Conjecture 2.15, which we recall statesthat, starting from an identifiable, strongly connected model with only one leak, removing theleak yields a model that is again identifiable. Here, we prove that (under some hypotheses)Conjecture 2.15 is equivalent to a new conjecture (Conjecture 3.6), which states that leakterms do not divide the singular-locus equation (Theorem 3.8). It will then follow thatConjecture 2.15 holds whenever Conjecture 3.6 is true (Remark 3.10).
Conjecture 3.6 (Leaks are not dividing terms) . Let f M = ( G, In, Out, Leak ) be a stronglyconnected linear compartmental model with at least one input and at least one leak. Assumethat f M is generically locally identifiable, and has singular-locus equation f . Then k ℓ ∤ f forall ℓ ∈ Leak . The following definition captures when a model has the same number of parameters ascoefficients of the input-output equation, that is, the resulting Jacobian matrix is square.
DENTIFIABILITY OF LINEAR COMPARTMENTAL MODELS 11
Definition 3.7.
Let M = ( G, In, Out, Leak ) be a strongly connected linear compartmentalmodel with | In | = | Out | = 1. The model M is square-Jacobian if the sum | E | + | Leak | equals the total number of nontrivial left-hand side and right-hand side coefficients assertedin Conjecture 2.8. Theorem 3.8 (Equivalence of Conjectures 2.15 and 3.6) . Let f M = ( G, In, Out, Leak ) bea strongly connected linear compartmental model with | In | = | Out | = | Leak | = 1 . Assumethat f M is generically locally identifiable and square-Jacobian. Assume that Conjecture 2.8holds for f M . Then, Conjecture 2.15 holds for f M if and only if Conjecture 3.6 holds for f M .Proof. Let ℓ denote the unique leak compartment. By Conjecture 2.8 and the square-Jacobian hypothesis, the number of nontrivial coefficients of the input-output equation (2)for f M is r = | E | + | Leak | . We denote these coefficients by e c , e c , . . . , e c r , where we havechosen some ordering so that e c r is the constant term of the left-hand side of (2).By Lemma 2.9, the corresponding coefficients of the input-output equation for the model M (obtained by removing the leak from M ′ ) are c i = e c i | k ℓ =0 , (5)for i = 1 , , . . . , r . We also know that c r = 0 (see [6, Remark 2.1]).Let J be the Jacobian matrix of ( c , c , . . . , c r − ), with respect to some ordering of the | E | parameters of M . Let e J denote the Jacobian matrix of ( e c , e c , . . . , e c r − ) with respect to thesame ordering of parameters, plus (for the last column of the matrix) the leak parameter k ℓ .We claim that the matrices J and e J (which are square by the square-Jacobian hypothesis)are related as follows: e J | k ℓ =0 = ∗ J ... ∗ . . . ∂ e c r ∂k ℓ (6)To prove the claim, we begin by noting that equation (5) and the equality c r = 0 implythat the matrix on the left-hand side of (6) and the matrix on the right-hand side have thesame columns, except possibly the last column. Next, the entry in the lower-right cornerof the matrix e J | k ℓ =0 is ∂ e c r ∂k ℓ | k ℓ =0 = ∂ e c r ∂k ℓ , where we are using the fact that k ℓ appears onlylinearly (and not to higher powers) in e c r . Hence, the claimed equality (6) holds, and so wecan take determinants to obtain: (cid:16) det e J (cid:17) | k ℓ =0 = ∂ e c r ∂k ℓ · (det J ) . (7)We next claim that ∂ e c r ∂k ℓ is nonzero. Indeed, this must hold in order for the inequality e c r = 0 and the equalities 0 = c r = e c r | k ℓ =0 to hold.Conjecture 3.6 holds for f M if and only if the left-hand side of (7) is nonzero. Also,Conjecture 2.15 holds for f M if and only if det J (appearing in the right-hand side of (7)) isnonzero. Thus, because ∂ e c r ∂k ℓ = 0, we see that the two conjectures are equivalent for f M . (cid:3) Remark 3.9 (Square vs. non-square Jacobian) . Theorem 3.8 pertains to square-Jacobianmodels. For non-square-Jacobian models that are identifiable, there are more coefficients of the input-output equation (2) than parameters, and being identifiable means that at leastone minor of the Jacobian matrix is nonzero. If such a minor comes from a submatrixcontaining the row corresponding to the constant term of the left-hand side of the input-output equation (i.e., the coefficient e c r in the proof of Theorem 3.8), then the proof easilygeneralizes to accommodate this case. On the other hand, when no such minor exists, we donot know how to address this scenario. However, we have never observed such a model! Remark 3.10.
By Theorem 3.8, Conjecture 3.6 is true whenever Conjecture 2.15 holds, forinstance, in the cases listed after Conjecture 2.15 (which include certain catenary, cycle, andmammillary models). 4.
Results on dividing edges
In this section, we address Conjecture 2.20, which we recall states that removing a dividingedge from an identifiable model results in a model that, if strongly connected, is unidentifi-able. Here we prove a special case of this conjecture, which pertains to the square-Jacobiancase (Theorem 4.1), and then investigate the non-square-Jacobian case (Section 4.2).4.1.
When distance from input to output increases.
We have observed that, for somemodels, a dividing edge appears in the shortest path from the input compartment to theoutput compartment and its removal increases the length of such a path. We can thereforeapply the conjectured formula for the number of coefficients to obtain the following result.
Theorem 4.1 (Remove dividing edge) . Let M be a strongly connected linear compartmentalmodel with | In | = | Out | = 1 and Leak = ∅ . Assume that M is square-Jacobian andgenerically locally identifiable. Assume, moreover, that k ij is a dividing edge such that themodel M ′ obtained from M by removing the edge k ij is strongly connected and the length ofthe shortest path from input to output has increased by at least . If Conjecture 2.8 holds forboth M and M ′ , then M ′ is unidentifiable.Proof. We know that In = Out , as the length of the shortest path from input to outputincreases when k ij is removed. For M , the number of coefficients equals the number ofparameters, namely, | E | + | Leak | . As for M ′ , the number of parameters is | E | + | Leak | − | E | + | Leak | −
2. Thus, as there are more parameters than coefficients of the input-output equation, M ′ is unidentifiable (by Proposition 2.11). (cid:3) Example 4.2.
Figure 5 depicts an identifiable, strongly connected, square-Jacobian linearcompartmental model M with dividing edge k . The model has 8 parameters, and theshortest path from input to output has length 1 (the edge k ). When the dividing edge k is removed, the resulting model M ′ is also strongly connected, but the length of the shortestpath from input to output has increased to 3 (the edges of this path are k , k , k ). Thus,by Theorem 4.1, M ′ is unidentifiable.The model in the following example is not covered by Theorem 4.1. Example 4.3.
We revisit the linear compartmental model from Figure 3 (in Section 2),which is strongly connected and square-Jacobian. The dividing edges are k , k , and k .The edge k is an interesting case of a dividing edge, in that the removal of k results ina model M ′ that is still strongly connected and the length of the shortest path from input DENTIFIABILITY OF LINEAR COMPARTMENTAL MODELS 13 k k k k k k k k in Figure 5.
Model with In = { } , Out = { } , and dividing edge k (in blue).to output has not increased (so, Theorem 4.1 does not apply). Nevertheless, consistent withConjecture 2.20, this model M ′ is unidentifiable: there are 8 coefficients and 7 parameters,and all minors of the 8 × k and k , removing either one yields a model that is not strongly connected and so is notconsidered by Conjecture 2.20.We end this subsection by returning to a model from Section 2. Example 4.4.
Recall that the linear compartmental model shown in Figure 2 is stronglyconnected and square-Jacobian (6 × k and k . Removing theedge k yields a model that is no longer strongly connected, so Conjecture 2.20 does notapply. Removing the edge k yields a strongly connected model that is unidentifiable, whichis consistent with Conjecture 2.20. This unidentifiability is explained by Theorem 4.1: thelength of the shortest path from input to output increases from 1 to 3.4.2. Non-square Jacobian matrices.
We end this section by briefly considering the casewhen an identifiable model has a non-square Jacobian matrix and therefore the singularlocus is defined by the set of all maximal minors of the Jacobian matrix. In this case, wepropose to extend Conjecture 2.20 as follows: If an edge divides every maximal minor of theJacobian matrix, then removing this edge results in a model that is unidentifiable.It is also natural to consider the possible, stronger conjecture that if an edge divides atleast one maximal minor of the Jacobian matrix, then removing this edge makes the modelunidentifiable. However, this conjecture is false, as the next example shows. Indeed, morework remains to be done in order to understand the information contained – both collectivelyand individually – in the minors of the Jacobian matrix.
Example 4.5.
Depicted in Figure 6 is a strongly connected linear compartmental modelwith 5 parameters and 6 coefficients, which results in a 6 × M = − ( k k − k − k k k + k k − k k k + k k − k k k − k k + k k k )( k − k ) ,M = − ( k k − k k − k k k + k k − k k k + 3 k k k − k k − k k k + k k − k k k + k k − k k k + 2 k k k k − k k k )( k − k ) ,
32 1 4 k k k k k in Figure 6.
Model with In = Out = { } . M = − ( k k − k k − k k k + k k k − k k k + 3 k k k − k k k + 2 k k k k − k k k k + k k k + k k − k k k + 3 k k k − k k − k k k + 2 k k k k − k k k + k k − k k k + k k − k k k + 2 k k k k − k k k k − k k k + 2 k k k k − k k k )( k − k ) ,M = − ( k − k )( k − k )( k − k ) k ,M = − ( k k − k k + k k )( k − k )( k − k )( k − k ) ,M = − ( k k − k k + k k k − k k + k k )( k − k )( k − k )( k − k ) . Notice that no edge parameter k ij divides all 6 of these minors. However, the edge k divides one of the 6 minors, M . When this edge is removed, the resulting model M ′ isno longer strongly connected, but by thinking of the edge k as a “pseudo-leak”, we cannevertheless view the model M ′ as a (strongly connected) cycle model on three nodes withinput, output, and leak in a single compartment – and so M ′ is identifiable [10].5. Discussion
This article was motivated by the following questions about linear compartmental models:
Question 5.1. (1) (When) does adding or removing an edge or a leak preserve identifiability?(2) (When) do edge or leak terms divide the singular-locus equation?(3) How do the above two questions interact?Some conjectured (partial) answers are as follows. Removing a leak preserves identifiability(Conjecture 2.15), removing a dividing edge never preserves identifiability (Conjecture 2.20),and leak terms do not divide the singular-locus equation (Conjecture 3.6). We proved sev-eral subcases of these conjectures and proved (under some hypotheses) the equivalence ofConjectures 2.15 and 3.6.Crucially, our proofs rely on a conjectured number of nontrivial coefficients of input-outputequations. Even if this conjecture is soon resolved, our work leaves open many cases whena model is unidentifiable but nevertheless has at least as many coefficients as parameters.This is an interesting future direction.More generally, we hope that our results inspire more answers to Question 5.1, which inturn will contribute to our ability to read important information about a model directlyfrom its structure. Indeed, it would be spectacular to be able to infer immediately from the
DENTIFIABILITY OF LINEAR COMPARTMENTAL MODELS 15 underlying combinatorics of a model which edges are dividing edges or even whether or notthe model is identifiable.
Acknowledgements.
Patrick Chan, Katherine Johnston, and Clare Spinner initiated thisresearch in the 2020 REU in the Department of Mathematics at Texas A&M University,supported by NSF grant DMS-1757872, in which Anne Shiu and Aleksandra Sobieska werementors. Anne Shiu was supported by NSF grant DMS-1752672, and acknowledges CashousBortner and Nicolette Meshkat for helpful discussions.
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