Mammillary Models in Compartmental Analysis

• Mammillary models are linear compartmental systems characterized by a central compartment exchanging material bidirectionally with peripheral compartments.
• They use explicit input–output formulas and combinatorial spanning forest techniques to reveal parameter identifiability and symmetry-induced ambiguities.
• The models support transfer-function realization in pharmacokinetics, facilitating physiologically interpretable parameter recovery via symmetric polynomial methods.

Mammillary models are linear compartmental models whose underlying directed graph is a star: one central compartment exchanges material bidirectionally with peripheral compartments, and there are no edges among peripherals. In the modern literature they are treated both as a distinguished subclass of bidirectional tree models and as a standard topology in pharmacokinetics, where the central compartment is often interpreted as plasma and peripheral compartments as tissue groups. Their mathematical theory now includes explicit input–output formulas, structural and parameter identifiability classifications, singular-locus descriptions, and transfer-function realization results (Gross et al., 2017, Bortner et al., 2021, Beltrami et al., 14 Jul 2025).

1. Graph structure and dynamical formulation

A linear compartmental model is a quadruple $(G,\mathrm{In},\mathrm{Out},\mathrm{Leak})$ in which $G=(V,E)$ is a directed graph, $\mathrm{In}$, $\mathrm{Out}$, and $\mathrm{Leak}$ are the input, output, and leak compartments, and each directed edge $j\to i$ carries a rate parameter. In one common notation, edges are labeled by $a_{ij}$ and leaks by $a_{0i}$; in another, edges are labeled by $k_{ij}$ and leaks by $k_{0i}$. The state equation is

$G=(V,E)$0

with $G=(V,E)$1 for $G=(V,E)$2. The compartmental matrix is determined by off-diagonal transfer rates and by diagonal entries equal to the negative sum of outgoing rates, together with leak terms where present (Gross et al., 2017, Bortner et al., 2021).

For mammillary models, the graph is the bidirected star $G=(V,E)$3: the center is compartment $G=(V,E)$4, the peripherals are $G=(V,E)$5, and the edge set is

$G=(V,E)$6

This is a bidirectional tree. Under the realization-oriented convention used in recent work, elimination occurs only from the central compartment, the input is an infusion into the central compartment, and the output is the central amount or concentration. The corresponding state-space matrices are

$G=(V,E)$7

with $G=(V,E)$8, $G=(V,E)$9, and $\mathrm{In}$0 for $\mathrm{In}$1 (Beltrami et al., 14 Jul 2025).

In the identifiability literature, mammillary models are studied under several placement conventions for input, output, and leak. A particularly important setting is the single-input/single-output case with $\mathrm{In}$2, in which the central compartment carries the input, output, and unique leak, while peripheral flows are parameterized by $\mathrm{In}$3 and $\mathrm{In}$4 (Gross et al., 2017).

2. Input–output equations and combinatorial coefficient formulas

The basic input–output relation for strongly connected models with one input in compartment $\mathrm{In}$5 and one output in compartment $\mathrm{In}$6 is

$\mathrm{In}$7

where $\mathrm{In}$8 and $\mathrm{In}$9 is the submatrix obtained by deleting row $\mathrm{Out}$0 and column $\mathrm{Out}$1. Meshkat–Sullivant–Eisenberg established this determinant formula, and subsequent work used it as the main gateway from state-space dynamics to algebraic identifiability (Gross et al., 2017).

A major development is the coefficient formula in terms of spanning incoming forests. Let $\mathrm{Out}$2 be the leak-augmented graph obtained by adjoining a node $\mathrm{Out}$3 and adding leak edges, and let $\mathrm{Out}$4 or $\mathrm{Out}$5 denote the auxiliary graphs used for the right-hand side coefficients after modifying the output compartment. If $\mathrm{Out}$6 is the product of edge labels in an incoming forest $\mathrm{Out}$7, then the coefficients of the input–output equation are sums of $\mathrm{Out}$8 over appropriate sets of forests. For one input and one output, possibly in distinct compartments,

$\mathrm{Out}$9

and, in the central-input/central-output mammillary case with $\mathrm{Leak}$0,

$\mathrm{Leak}$1

These formulas replace determinant expansion by a graph-theoretic enumeration problem (Bortner et al., 2021, Gross et al., 2017).

For mammillary models, the forest formulas collapse to explicit expressions in elementary symmetric polynomials. Writing $\mathrm{Leak}$2 for the $\mathrm{Leak}$3-th elementary symmetric polynomial and $\mathrm{Leak}$4 for the same polynomial with $\mathrm{Leak}$5 omitted,

$\mathrm{Leak}$6

for $\mathrm{Leak}$7, and

$\mathrm{Leak}$8

for $\mathrm{Leak}$9. These formulas make the symmetry of the central-to-peripheral outflow rates explicit: the $j\to i$0 enter symmetrically, whereas the inflows $j\to i$1 and the leak $j\to i$2 enter linearly (Gross et al., 2017).

A complementary derivation specialized to the hub-and-spoke topology uses a Schur complement. With $j\to i$3, $j\to i$4 the total outflow from the center including a possible central leak, and $j\to i$5 the total outflow from peripheral $j\to i$6, one obtains

$j\to i$7

so the right-hand side coefficients are exactly the elementary symmetric polynomials in the peripheral diagonal terms $j\to i$8 (Gerberding et al., 2019).

3. Structural identifiability and the role of input, output, and leaks

For strongly connected one-input/one-output models, generic local identifiability is equivalent to a rank condition on the Jacobian of the coefficient map. If $j\to i$9 denotes the Jacobian of the map from parameters to non-monic input–output coefficients, then the model is generically locally identifiable if and only if $a_{ij}$0 at a generic parameter point. Equivalently, the generic fibers of the coefficient map are finite (Gross et al., 2017, Gerberding et al., 2019).

For bidirectional tree models, Bortner et al. obtained a complete visual classification. Let $a_{ij}$1 be a bidirectional tree model with exactly one input and one output. Then $a_{ij}$2 is generically locally identifiable if and only if

$a_{ij}$3

In a mammillary graph this specializes to a concise criterion: a model $a_{ij}$4 with $a_{ij}$5 is generically locally identifiable if and only if $a_{ij}$6 and one of the following holds: $a_{ij}$7, $a_{ij}$8, or $a_{ij}$9. Distinct peripheral input and output compartments are at graph distance $a_{0i}$0 via the hub and are therefore not structurally identifiable (Bortner et al., 2021).

A more specialized leak result was established earlier for the standard mammillary placement with $a_{0i}$1. If a catenary, cycle, or mammillary model has exactly one input and exactly one output, both in the first compartment, and exactly one leak, then the one-leak model is generically locally identifiable and so is the model obtained by removing the leak. Together with the general operation that adding one leak preserves identifiability, this implies that moving or deleting a single leak preserves identifiability in this mammillary setting (Gerberding et al., 2019).

These results separate two distinct issues. The tree classification determines when every parameter is generically locally identifiable, while the leak-moving theorem shows that, for the standard central-input/central-output mammillary configuration, the presence or location of a single leak does not destroy generic local identifiability (Bortner et al., 2021, Gerberding et al., 2019).

4. Singular locus, permutation symmetry, and identifiability degree

Beyond generic local identifiability, one can ask for the parameter values at which the Jacobian drops rank. This algebraic subset is the singular locus. In cases where it has codimension $a_{0i}$2, it is defined by a single singular-locus equation. For mammillary models with $a_{0i}$3, the singular-locus equation is

$a_{0i}$4

Accordingly, local non-identifiability occurs precisely when some central-to-peripheral outflow $a_{0i}$5 is zero or when two such outflow rates coincide. The squared Vandermonde factor records the stronger degeneracy associated with equality of outflow rates (Gross et al., 2017).

The small-dimensional examples are explicit. For $a_{0i}$6, with parameters $a_{0i}$7, the determinant of the Jacobian of $a_{0i}$8 is

$a_{0i}$9

so rank drops exactly when $k_{ij}$0, $k_{ij}$1, or $k_{ij}$2. For $k_{ij}$3,

$k_{ij}$4

and equality of any two outflow rates makes the corresponding peripheral compartments indistinguishable from input–output data (Gross et al., 2017).

The same symmetry governs the identifiability degree, defined as the number of parameter values mapping to a generic input–output data vector. For mammillary models with $k_{ij}$5,

$k_{ij}$6

This is the permutation ambiguity of the peripherals: the coefficient map depends on the multiset of outflows through symmetric polynomials, not on the labels of the peripheral compartments. Cobelli, Lepschy, and Romanin Jacur had previously computed this degree for mammillary models, and the later singular-locus analysis places it in a broader algebraic framework (Gross et al., 2017).

The singular-locus equation also controls certain submodel questions. If a generically locally identifiable model has singular-locus equation $k_{ij}$7, and a set of edges is deleted so that strong connectivity is preserved and $k_{ij}$8 is not annihilated by setting the deleted parameters to zero, then the resulting submodel remains generically locally identifiable. In mammillary models, however, deleting an outflow or an inflow typically breaks the hypotheses: outflow deletions force a factor of $k_{ij}$9 to vanish, and inflow deletions destroy strong connectivity (Gross et al., 2017).

5. Parameter-level identifiability in the five mammillary families

Recent work refined structural identifiability to parameter-level global versus local identifiability for leak-free mammillary models with one input and one output. Up to relabeling of peripherals, there are five infinite families: $k_{0i}$0, $k_{0i}$1, $k_{0i}$2, $k_{0i}$3, and $k_{0i}$4, where the ordered pair indicates the locations of input and output at the center or at peripheral compartments (Clemens et al., 27 Jun 2025).

Family	Placement	Main conclusion
$k_{0i}$5	center to center	all parameters are SLING
$k_{0i}$6	center to peripheral	$k_{0i}$7 globally identifiable; others SLING
$k_{0i}$8	peripheral to center	$k_{0i}$9 globally identifiable; $G=(V,E)$00 generically globally identifiable; others SLING
$G=(V,E)$01	same peripheral to itself	$G=(V,E)$02 and $G=(V,E)$03 generically globally identifiable; others SLING
$G=(V,E)$04	peripheral to different peripheral	$G=(V,E)$05 for $G=(V,E)$06 are SLING; remaining parameters conjecturally unidentifiable for $G=(V,E)$07

A key lemma applies across all five families: in any strongly connected leak-free model with one input $G=(V,E)$08 and one output $G=(V,E)$09, if the edge $G=(V,E)$10 is present, then the corresponding parameter $G=(V,E)$11 equals the coefficient $G=(V,E)$12 in the input–output equation and is therefore globally identifiable. In mammillary models this immediately identifies the direct edge from input to output whenever that edge exists (Clemens et al., 27 Jun 2025).

The family $G=(V,E)$13 is especially explicit. Writing $G=(V,E)$14, one has

$G=(V,E)$15

Hence $G=(V,E)$16, and the remaining center–peripheral return rate adjacent to the input/output pair is recovered by

$G=(V,E)$17

with denominator nonzero generically (Clemens et al., 27 Jun 2025).

For $G=(V,E)$18, the two rates on the edge pair connecting the distinguished peripheral to the center are isolated by simple coefficient differences:

$G=(V,E)$19

and

$G=(V,E)$20

All remaining parameters are only structurally locally identifiable but not globally identifiable, abbreviated SLING, because of residual peripheral permutation symmetry (Clemens et al., 27 Jun 2025).

The family $G=(V,E)$21 illustrates a different phenomenon. Here $G=(V,E)$22 is globally identifiable, but $G=(V,E)$23 is only generically locally identifiable: it satisfies an explicit polynomial equation of degree $G=(V,E)$24 in terms of the input–output coefficients. In $G=(V,E)$25, the coefficients determine the elementary symmetric polynomials of the rates $G=(V,E)$26, so these parameters are recoverable only up to permutation. For $G=(V,E)$27, the paper states the conjecture that the remaining parameters are unidentifiable (Clemens et al., 27 Jun 2025).

6. Transfer-function realization and pharmacokinetic interpretation

A separate but closely related line of work asks when a given transfer function can be realized by a mammillary compartmental system. For a strictly proper SISO transfer function

$G=(V,E)$28

of order $G=(V,E)$29, the mammillary realization with central input and central output has canonical transfer form

$G=(V,E)$30

There exists a unique parameter vector $G=(V,E)$31 such that

$G=(V,E)$32

with $G=(V,E)$33 in mammillary form if and only if $G=(V,E)$34 has relative degree $G=(V,E)$35 and the numerator $G=(V,E)$36 is monic with simple, real, and nonzero roots (Beltrami et al., 14 Jul 2025).

The positive-realization theorem strengthens these conditions. There exists a unique strictly positive mammillary parameter vector if and only if: $G=(V,E)$37 has relative degree $G=(V,E)$38; $G=(V,E)$39 is monic with simple, real, strictly negative roots; $G=(V,E)$40; and, for each zero $G=(V,E)$41 of $G=(V,E)$42,

$G=(V,E)$43

The constructive recovery is

$G=(V,E)$44

Under the ordering assumption $G=(V,E)$45, this realization is unique (Beltrami et al., 14 Jul 2025).

These realizability results are motivated by pharmacokinetics. In the three-compartment case,

$G=(V,E)$46

Here $G=(V,E)$47 is central, $G=(V,E)$48 is a fast peripheral compartment, and $G=(V,E)$49 is a slow peripheral compartment. The realization theorem provides a direct map from an identified transfer function to these physiologically interpretable rates (Beltrami et al., 14 Jul 2025).

The same paper develops a propofol application by coupling a three-compartment mammillary pharmacokinetic model to a first-order effect-site pharmacodynamic compartment and a Hill nonlinearity. For a 40-year-old, 163 cm, 54 kg female patient, parameters derived from the Schnider model were recovered by the realization algorithm, up to permutation of the two peripheral subsystems. This application shows how mammillary constraints restrict the admissible pole-zero structure and thereby enforce physiological interpretability (Beltrami et al., 14 Jul 2025).

An additional structural property follows from positivity. If $G=(V,E)$50 and all $G=(V,E)$51, then $G=(V,E)$52 is a Metzler compartmental matrix whose eigenvalues are real and nonpositive. The proof uses a diagonal scaling that makes $G=(V,E)$53 similar to a symmetric matrix, so mammillary realizations combine compartmental positivity with real decay modes (Beltrami et al., 14 Jul 2025).

7. Assumptions, scope, and open problems

Most rigorous results on mammillary models are derived under restrictive but transparent assumptions. Structural identifiability theorems for bidirectional trees assume exactly one input and one output, generic initial conditions, and at most one leak. Parameter-level global/local identifiability results for the five mammillary families assume no leaks. Realization results assume central input and central output, elimination only from the central compartment, and distinct ordered peripheral return rates (Bortner et al., 2021, Clemens et al., 27 Jun 2025, Beltrami et al., 14 Jul 2025).

Strong connectivity is essential in the Jacobian-rank criterion and in the tree classification. Full bidirectional hub–peripheral connectivity guarantees it for the complete mammillary graph, but many edge-deletion operations destroy it. In the singular-locus framework, preserving identifiability under edge deletion requires both strong connectivity and the condition that the deleted parameters do not annihilate the singular-locus polynomial; mammillary submodels often fail one or both requirements (Gross et al., 2017).

Genericity assumptions matter. Equal peripheral outflow rates place a central-input/central-output mammillary model on the singular locus, where the Vandermonde factor vanishes. This is the algebraic expression of a symmetry-induced ambiguity: when peripheral rates coincide, the model loses the ability to distinguish the corresponding compartments from input–output data. A plausible implication is that practical estimation procedures should avoid imposing equality constraints unless such symmetries are intended (Gross et al., 2017).

Several open directions are already explicit in the literature. One line concerns catenary models, for which a full singular-locus formula was conjectured but not proved in the 2017 analysis. Another concerns the family $G=(V,E)$54, where parameter-level unidentifiability of several edges is conjectured for $G=(V,E)$55. Further extensions to models with leaks, multiple inputs or outputs, and non-tree topologies remain outside the scope of the current complete classifications (Gross et al., 2017, Clemens et al., 27 Jun 2025, Bortner et al., 2021).

Taken together, these results show that mammillary models occupy a mathematically unusual position. They are simple enough to admit closed-form coefficient formulas, singular-locus equations, and transfer-function realization criteria, yet rich enough to exhibit factorial identifiability degree, extensive SLING behavior, and sharp dependence on input–output placement. That combination explains their continued role as benchmark objects in compartmental-model theory and as physiologically interpretable models in pharmacokinetics (Gross et al., 2017, Clemens et al., 27 Jun 2025, Beltrami et al., 14 Jul 2025).