Identifiability of linear compartment models:
the singular locus
This work addresses the problem of identifiability, that is, the question of whether parameters can be recovered from data, for linear compartment models. Using standard differential algebra techniques, the question of whether a given model is generically locally identifiable is equivalent to asking whether the Jacobian matrix of a certain coefficient map, arising from input-output equations, is generically full rank. We give a formula for these coefficient maps in terms of acyclic subgraphs of the model’s underlying directed graph. As an application, we prove that two families of linear compartment models, cycle and mammillary (star) models with input and output in a single compartment, are identifiable, by determining the defining equation for the locus of non-identifiable parameter values. We also state a conjecture for the corresponding equation for a third family: catenary (path) models. These singular-locus equations, we show, give information on which submodels are identifiable. Finally, we introduce the identifiability degree, which is the number of parameter values that match generic input-output data. This degree was previously computed for mammillary and catenary models, and here we determine this degree for cycle models. Our work helps shed light on the question of which linear compartment models are identifiable.
This work focuses on the identifiability problem for linear compartment models. Linear compartment models are used extensively in biological applications, such as pharmacokinetics, toxicology, cell biology, physiology, and ecology [2, 3, 7, 9, 12]. Indeed, these models are now ubiquitous in pharmacokinetics, with most kinetic parameters for drugs (half-lives, residence times, and so on) based at least in part on linear compartment model theory [13, 18].
A mathematical model is identifiable if its parameters can be recovered from data. Using standard differential algebra techniques, the question of whether a given linear compartment model is (generically locally) identifiable is equivalent to asking whether the Jacobian matrix of a certain coefficient map (arising from certain input-output equations) is generically full rank. Recently, Meshkat, Sullivant, and Eisenberg gave a general formula for the input-output equations of a linear compartment model . Using this formula, it is easy to check whether a given model is (generically locally) identifiable. Nevertheless, we would like to bypass this formula. That is, can we determine whether a model is identifiable by simply inspecting its underlying directed graph? Or, as a first step, if a model is identifiable, when can we conclude that a given submodel is too? Our interest in submodels, obtained by removing edges, has potential applications: the operation of removing an edge in a model may correspond to a biological intervention, such as a genetic knockout or a drug that inhibits some activity. Identifiable submodels have been studied by Vajda and others [14, 16].
This work begins to answer the questions mentioned above. Our first main result is a formula for the coefficient map in terms of forests (acyclic subgraphs) in the directed graph associated to the model (Theorem 4.5). Our second result gives information on which edges of a model can be deleted while still preserving identifiability (Theorem 3.1). Our remaining results pertain to three well-known families of linear compartment models, which are depicted in Figures 1 and 2: catenary (path graph) models, mammillary (star graph) models, and cycle models .
a defining equation for the set of non-identifiable parameter values (the singular locus), and
the identifiability degree: this degree is if exactly sets of parameter values match generic input-output data.
|Model||Equation of singular locus||Identifiability|
We are, to the best of our knowledge, the first to study this singular locus.
The outline of our work is as follows. In Section 2, we introduce linear compartment models and define the singular locus. In Section 3, we prove our result on how the singular locus gives information on identifiable submodels. In Section 4, we give a new combinatorial formula for the coefficients of the input-output equations for linear compartment models with input and output in a single compartment. We use this formula to prove, in Sections 5 and 6, the results on the singular-locus equations and identifiability degrees mentioned above for the models in Figures 1 and 2. We conclude with a discussion in Section 7.
In this section, we recall linear compartment models, their input-output equations, and the concept of identifiability. We also introduce the main focus our work: the locus of non-identifiable parameter values and the equation that defines it.
2.1. Linear compartment models
A linear compartment model (alternatively, linear compartmental model) consists of a directed graph together with three sets . Each vertex corresponds to a compartment in the model and each edge corresponds to a direct flow of material from the -th compartment to the -th compartment. The sets are the sets of input compartments, output compartments, and leak compartments, respectively.
Following the literature, we will indicate output compartments by this symbol: . Input compartments are labeled by “in”, and leaks are indicated by outgoing edges. For instance, each of the linear compartment models depicted in Figures 1 and 2 have .
To each edge of , we associate a parameter , the rate of flow from compartment to compartment . To each leak node , we associate a parameter , the rate of flow from compartment leaving the system. Let . The compartmental matrix of a linear compartment model is the matrix with entries given by:
A linear compartment model defines a system of linear ODEs (with inputs ) and outputs as follows:
where for .
We now define the concepts of strongly connected and inductively strongly connected.
A directed graph is strongly connected if there exists a directed path from each vertex to every other vertex. A directed graph is inductively strongly connected with respect to vertex if there is an ordering of the vertices that starts at vertex such that each of the induced subgraphs is strongly connected for .
A linear compartment model is strongly connected (respectively, inductively strongly connected) if is strongly connected (respectively, inductively strongly connected).
The two most common classes of compartmental models are mammillary (star) and catenary (path) model structures (see Figures 1 and 2). Mammillary models consist of a central compartment surrounded by and connected with peripheral (noncentral) compartments, none of which are connected to each other . Catenary models have all compartments arranged in a chain, with each connected (in series) only to its nearest neighbors . In a typical pharmacokinetic application, the central compartment of a mammillary model consists of plasma and highly perfused tissues in which a drug distributes rapidly. For catenary models, the drug distributes more slowly. For examples of how mammillary and catenary models are used in practice, see [7, 9, 17].
2.2. Input-output equations
The input-output equations of a linear compartment model are equations that hold along any solution of the ODEs (1), and which involve only the parameters , input variables , output variables , and their derivatives. The general form of these equations was given by Meshkat, Sullivant, and Eisenberg [11, Corollary 1]. The version of their result we state here is for the case of a single input/output compartment:
Proposition 2.2 (Meshkat, Sullivant, and Eisenberg).
Consider a linear compartment model that is strongly connected, has an input and output in compartment 1 (and no other inputs or outputs), and has at least one leak. Let denote the compartmental matrix, let be the differential operator , and let denote the submatrix of obtained by removing row 1 and column 1. Then the input-output equation (of lowest degree) is the following:
Consider the following catenary model (the case from Figure 1):
By Proposition 2.2, the input-output equation is:
which, when expanded, becomes:
Observe, from the left-hand side of this equation, that the coefficient of corresponds to the set of forests (acyclic subgraphs) of the model that have edges and at most 1 outgoing edge per compartment. As for the right-hand side, the coefficient of corresponds to similar -edge forests in the following model:
This combinatorial interpretation of the coefficients of the input-output equation generalizes, as we will see in Theorem 4.5.
A linear compartment model is generically structurally identifiable if from a generic choice of the inputs and initial conditions, the parameters of the model can be recovered from exact measurements of both the inputs and the outputs. We now define this concept precisely.
Let be a linear compartment model. The coefficient map is the function that is the vector of all coefficient functions of the input-output equation (here is the total number of coefficients). Then is:
globally identifiable if is one-to-one, and is generically globally identifiable if is one-to-one outside a set of measure zero.
locally identifiable if around every point in there is an open neighborhood such that is one-to-one, and is generically locally identifiable if, outside a set of measure zero, every point in has such an open neighborhood .
unidentifiable if is infinite-to-one.
Since the coefficients in are all polynomial functions of the parameters, the model is generically locally identifiable if and only if the image of has dimension equal to the number of parameters, i.e., . The dimension of the image of a map is equal to the rank of the Jacobian matrix at a generic point. Thus we have the following result, which is [11, Proposition 2]:
Proposition 2.5 (Meshkat, Sullivant, and Eisenberg).
A linear compartment model is generically locally identifiable if and only if the Jacobian matrix of its coefficient map , when evaluated at a generic point, is equal to .
As an alternative to Proposition 2.5, one can test identifiability by using a Gröbner basis to solve the system of equations , for some choice of algebraically independent . The model is then globally identifiable if there is a unique solution for in terms of , locally identifiable if there are a finite number of solutions for in terms of , and unidentifiable if there are an infinite number of solutions. In practice, Gröbner basis computations are more computationally expensive than Jacobian calculations, thus we will test local identifiability using the Jacobian test (Proposition 2.5).
We now examine when the Jacobian is generically full rank, but certain parameter choices lead to rank-deficiency. We call parameter values that lead to this rank-deficiency non-identifiable. Note that the parameters of these models are generically identifiable, and in the identifiability literature are called “identifiable” , but for our purposes, we are examining the non-generic case and thus denote the values of these parameters “non-identifiable”.
Let be a linear compartment model that is generically locally identifiable. Let denote the coefficient map. The locus of non-identifiable parameter values, or, for short, the singular locus is the subset of the parameter space where the Jacobian matrix of has rank strictly less than .
Thus, the singular locus is the defined by the set of all minors of . We will focus on the cases when only a single such minor, which we give a name to below, defines the singular locus:
Let be a linear compartment model, with coefficient map . Suppose is generically locally identifiable (so, ).
If (the number of parameters equals the number of coefficients), then is the equation of the singular locus.
Assume . Suppose there is a choice of coefficients from , with the resulting restricted coefficient map, such that if and only if the Jacobian of has rank strictly less than . Then is the equation of the singular locus.
The equation of the singular locus, when , is defined only up to sign, as we do not specify the order of the coefficients in . When , there need not be a single minor that defines the singular locus, and thus a singular-locus equation as defined above might not exist. We, however, have not encountered such a model, although we suspect one exists. Accordingly, we ask, is there always a choice of coefficients or, equivalently, rows of , such that this square submatrix is rank-deficient if and only if the original matrix is? And, when such a choice exists, is this choice of coefficients unique?
In applications, we typically are only interested in the factors of the singular-locus equation: we only care whether, e.g., divides the equation (i.e., whether is non-identifiable) and not which higher powers , for positive integers , also divide it.
One aim of our work is to investigate the equation of the singular locus for mammillary, catenary, and cycle models with a single input, output, and leak in the first compartment. As a start, all of these families of models are (at least) generically locally identifiable:
Catenary and mammillary models are inductively strongly connected with edges. T hus, catenary and mammillary models with a single input and output in the first compartment and leaks from every compartment have coefficient maps with images of maximal dimension [10, Theorem 5.13]. Removing all the leaks except one from the first compartment, we can apply [11, Theorem 1] and obtain generic local identifiability.
3. The singular locus and identifiable submodels
One reason a model’s singular locus is of interest is because it gives us information regarding the identifiability of particular parameter values. Indeed, for generically locally identifiable models, the singular locus is the set of parameter values that cannot be recovered, even locally. A second reason for studying the singular locus, which is the main focus of this section, is that the singular-locus equation gives information about which submodels are identifiable.
Theorem 3.1 (Identifiable submodels).
Let be a linear compartment model that is strongly connected and generically locally identifiable, with singular-locus equation . Let be the model obtained from by deleting a set of edges of . If is strongly connected, and is not in the ideal (or, equivalently, after evaluating at for all , the resulting polynomial is nonzero), then is generically locally identifiable.
Let , the submodel , the polynomial , and the ideal be as in the statement of the theorem. Thus, the following polynomial , obtained by evaluating at for all deleted edges in , is not the zero polynomial:
Let . Let denote the matrix obtained from by setting for all . The determinant of is the nonzero polynomial , so is full rank when evaluated at any parameter vector outside the measure-zero set . (Here, denotes the real vanishing set of .) Thus, the matrix obtained from by deleting the set of columns corresponding to , is also full rank () outside of .
Choose rows of that are linearly independent outside some measure-zero set in . (Such a choice exists, because, otherwise, would be rank-deficient on all of and thus so would the generically full-rank matrix , which is a contradiction.) These rows form an matrix that we call .
Let be obtained from by restricting to the coordinates corresponding to the above choice of rows of , and also setting for all . By construction and by Proposition 2.2 (here we use that is strongly connected), is a choice of coefficients from the input-output equations of , and, by construction, the Jacobian matrix of is (whose rows we chose to be generically full rank). Hence, is generically locally identifiable. ∎
Consider the following (strongly connected) linear compartment model :
This model is generically locally identifiable, and the equation of the singular locus is:
This equation is not divisible by , and the model obtained by removing that edge (labeled by ) is strongly connected. So, by Theorem 3.1, is generically locally identifiable.
The converse of Theorem 3.1 does not hold, as we see in the following example.
Example 3.3 (Counterexample to converse of Theorem 3.1).
Consider again the model from Example 3.2. The submodel obtained by deleting the edges labeled by and is generically locally identifiable (by Theorem 5.2: the submodel is the 4-compartment cycle model). Nevertheless, the singular-locus equation of is divisible by and thus the equation is in the ideal .
Example 3.3, our counterexample to the converse of Theorem 3.1, involved deleting two edges (). We do not know of a counterexample that deletes only one edge, and we end this section with the following question.
In the setting of Theorem 3.1, if a parameter divides , does it follow that the model obtained by deleting the edge labeled by is unidentifiable (assuming that is strongly connected)?
4. The coefficient map and its Jacobian matrix
Recall that for strongly connected linear compartment models with input and output in one compartment, plus at least one leak, the input-output equation was given in equation (2) (in Proposition 2.2). In this section, we give a new combinatorial formula for the coefficients of the input-output equation (Theorem 4.5).
To state Theorem 4.5, we must define some graphs associated to a model. In what follows, we use “graph” to mean “directed graph”.
Consider a linear compartment model with compartments.
The leak-augmented graph of , denoted by , is obtained from by adding a new node, labeled by 0, and adding edges labeled by , for every leak .
The graph , for some , is obtained from by completing these steps:
Delete compartment , by taking the induced subgraph of with vertices
, and then:
For each edge (with label ) in ,if (i.e, with label is an edge in ), then label the leak in by ; if, on the other hand, , then add to the edge with label .
Figure 3 displays a model , its leak-augmented graph , and the graphs and . The compartmental matrix of is:
The compartmental matrix that corresponds to is obtained from by removing row 1 and column 1. Similarly, for , the corresponding compartmental matrix comes from deleting row 2 and column 2 from . This observation generalizes (see Lemma 4.3).
Consider a linear compartment model with compartmental matrix and compartments. Let , for some , be as in Definition 4.1. Then for any model whose leak-augmented graph is , the compartmental matrix of is the matrix obtained from by removing row and column .
Let , , and be as in the statement of the lemma. Let denote the matrix obtained from by removing row and column . We must show that the compartmental matrix of equals . The graph is obtained by taking the induced subgraph of formed by all vertices except – which ensures that the off-diagonal entries of the compartmental matrix of equal those of – and then replacing edges directed toward with leak edges (and combining them as necessary with existing leak edges) – which ensures that the diagonal entries of the compartmental matrix also equal those of . Thus, is the compartmental matrix of . ∎
The following terminology matches that of Buslov :
Let be a (directed) graph.
A spanning subgraph of is a subgraph of with the same set of vertices as .
An incoming forest is a directed graph such that (a) the underlying undirected graph has no cycles and (b) each node has at most one outgoing edge.
For an incoming forest , let denote the product of the labels of all edges in the forest, that is, , where labels the edge .
Let denote the set of all -edge, spanning, incoming forests of .
4.2. A formula for the coefficient map
Our formula for the coefficient map expresses each coefficient as a sum, over certain spanning forests, of the product of the edge labels in the forest (Theorem 4.5). The formula is an “expanded out” version of a result of Meshkat and Sullivant [10, Theorem 3.2] that showed the coefficient map factors through the cycles in the leak-augmented graph. The difference is due to the fact that Meshkat and Sullivant treated diagonal entries of as separate variables (e.g., ), while our diagonal entries are negative sums of a leak and/or rates (e.g., ).
Theorem 4.5 (Coefficients of input-output equations).
Consider a linear compartment model that is strongly connected and has an input and output in compartment 1 (and no other inputs or outputs) and at least one leak. Let denote the number of compartments, and the compartmental matrix. Write the input-output equation (2) as:
Then the coefficients of this input-output equation are as follows:
The proof of Theorem 4.5 requires the following result, which interprets the coefficients of the characteristic polynomial of a compartmental matrix (with no assumptions on the model’s connectedness, number of leaks, etc.):
Let be the compartmental matrix of a linear compartment model with compartments and leak-augmented graph . Write the characteristic polynomial of as:
Then (for ) is the sum over -edge, spanning, incoming forests of , where each summand is the product of the edge labels in the forest:
In the Appendix, we prove Proposition 4.6 and explain how it is related to similar results.
Proof of Theorem 4.5.
By Proposition 2.2, the coefficient of in the input-output equation (3) is the coefficient of in the characteristic polynomial of the compartmental matrix . Hence, the desired result follows immediately from Proposition 4.6.
Now consider the right-hand side of the input-output equation (3). Let denote the matrix obtained from by removing row 1 and column 1. By Lemma 4.3, is the compartmental matrix for any model with leak-augmented graph . So, by Proposition 4.6, the sum equals the coefficient of in the characteristic polynomial (where the first identity matrix has size and the second has size ). This coefficient, by Proposition 2.2, equals , and this completes the proof. ∎
Remark 4.7 (Jacobian matrix of the coefficient map).
In the setting of Theorem 4.5, each coefficient of the input-output equation is the sum of products of edge labels of a forest, and thus is multilinear in the parameters . Therefore, in the row of the Jacobian matrix corresponding to , the entry in the column corresponding to some is obtained from by setting =1 in those terms divisible by and then setting all other terms to 0.
5. The singular locus: mammillary, catenary, and cycle models
In this section, we establish the singular-locus equations for the mammilary (star) and cycle models, which were displayed in Table 1 (Theorems 5.1 and 5.2). We also state our conjecture for the singular-locus equation for the catenary (path) model (Conjecture 5.3). We also pose a related conjecture for models that are formed by bidirectional trees, which include the catenary model (Conjecture 5.6).
5.1. Mammillary (star) models
Theorem 5.1 (Mammillary).
Assume . The -compartment mammillary (star) model in Figure 2 is generically locally identifiable, and the equation of the singular locus is:
The compartmental matrix for this model is
Let denote the -th elementary symmetric polynomial on ; and let denote the -th elementary symmetric polynomial on . Then, the coefficients on the left-hand side of the input-output equation (2) are, by Theorem 4.5, the following:
for . As for the coefficients of the right-hand side of the input-output equation, they are as follows, by Proposition 2.2:
Consider the coefficient map . Its Jacobian matrix, where the order of variables is , has the following form:
where is the following matrix:
Thus, to prove the desired formula (4), we need only show that the determinant of equals, up to sign, the Vandermonde polynomial on :
To see this, note first that both polynomials have the same multidegree: the degree with respect to the ’s of is (because the entries in row- of have degree ), which equals , and this is the degree of the Vandermonde polynomial on the right-hand side of equation (5). Also, note that both polynomials are, up to sign, monic.
So, to prove the claimed equality (5), it suffices to show that when , the term divides . Indeed, when , then the columns of that correspond to and (namely, the -st and -st columns) coincide, and thus . Hence, (by the Nullstellensatz). ∎
5.2. Cycle models
Theorem 5.2 (Cycle).
Assume . The -compartment cycle model in Figure 2 is generically locally identifiable, and the equation of the singular locus is:
The compartmental matrix for this model is
Let denote the -th elementary symmetric polynomial on
; and let denote the -th elementary symmetric polynomial on , , where . Then, the coefficients on the left-hand side of the input-output equation (3) are, by Theorem 4.5, the following:
As for the coefficients of the right-hand side of the input-output equation, they are as follows, by Proposition 2.2:
Consider the coefficient map . Its Jacobian matrix, where the order of variables is , has the following form:
where is the following matrix:
In the upper-left -block of the matrix in equation (9), rows 2 through are scalar multiples of the bottom row of 1’s. Thus, if we let denote the square matrix (of size ) obtained by removing from rows 2 through , then the singular-locus equation of the model is . Indeed, all nonzero minors of are scalar multiples of , and thus the singular locus is defined by the single equation .
5.3. Catenary (path) models
Assume . For the -compartment catenary (path) model in Figure 1, the equation of the singular locus is:
The structure of the conjectured equation (10) suggests a proof by induction on , but we currently do not know how to complete, for such a proof, the inductive step.
We can prove the following weaker version of Conjecture 5.3:
For the -compartment catenary (path) model in Figure 1, the following parameters divide the equation of the singular locus:
By Theorem 4.5, the coefficients of the input-output equation of the catenary model arise from spanning forests of the following graphs:
More specifically, some of the coefficients are as follows: