Mammillary Realizations in Compartmental Systems

• Mammillary realizations are structured linear-system models where a central compartment exchanges material with isolated peripheral compartments, preserving physiological topology.
• A unique realization exists when the transfer function has relative degree 1 and a monic numerator with simple, real nonzero roots, leading to explicit, constructive parameter-recovery formulas.
• This framework underpins applications such as pharmacokinetics, where it enables physiologically interpretable drug distribution models and aids in parameter identifiability analyses.

Searching arXiv for papers on mammillary realizations and adjacent realization theory. arXiv_search(query="mammillary realizations", max_results=10) Mammillary realizations are structured linear-system realizations by mammillary compartmental models, a subclass of compartmental systems in which one central compartment exchanges material with all peripheral compartments, peripheral compartments do not communicate with one another, the input and output are placed in prescribed compartments, and in the basic realization problem elimination occurs only from the central compartment (Beltrami et al., 14 Jul 2025). In contrast to arbitrary state-space realizations up to similarity, mammillary realizations preserve a fixed physiological topology and assign direct meanings to parameters such as transfer and elimination rates. Recent work has given an exact realizability theorem for a canonical continuous-time SISO mammillary form, constructive parameter-recovery formulas, a pharmacokinetic application to propofol, and a separate identifiability theory for leak-free one-input/one-output mammillary models with fixed graph structure (Beltrami et al., 14 Jul 2025, Clemens et al., 27 Jun 2025).

1. Canonical mammillary model and realization problem

The ambient system class is the continuous-time SISO linear system

$$\begin{cases} \dot x(t)=Ax(t)+Bu(t),\ y(t)=Cx(t). \end{cases}$$

A compartmental system satisfies

$b_i\ge 0,\qquad c_j\ge 0,$

$a_{ij}\ge 0\quad \text{for } i\neq j,$

$a_{ii}+\sum_{j\neq i} a_{ji}\le 0.$

These conditions encode mass conservation and positivity: states are amounts of substance, off-diagonal entries are inter-compartment flows, diagonal terms account for outflow or elimination, input injects material, and output measures nonnegative amounts (Beltrami et al., 14 Jul 2025).

Within this class, the mammillary model studied in the realization theorem has the form

$$A= \begin{bmatrix} -k_{10}-\sum_{i=2}^n k_{1i} & k_{21} & k_{31} & \cdots & k_{n1}\ k_{12} & -k_{21} & 0 & \cdots & 0\ k_{13} & 0 & -k_{31} & \cdots & 0\ \vdots & \vdots & \vdots & \ddots & \vdots\ k_{1n} & 0 & 0 & \cdots & -k_{n1} \end{bmatrix}, \qquad B=e_1=[1,0,\dots,0]^T,\qquad C=e_1^T=[1,0,\dots,0].$$

Here compartment $1$ is the central compartment, compartments $2,\dots,n$ are peripheral compartments, there are no peripheral-to-peripheral flows, each peripheral exchanges material only with the central compartment, input enters only the central compartment, output measures only the central compartment, and the only elimination parameter is $k_{10}$ (Beltrami et al., 14 Jul 2025).

The associated realization problem is: given a rational transfer function

$H(s)=\frac{\beta(s)}{\alpha(s)}$

of order $n$, determine when there exists a parameter vector

$b_i\ge 0,\qquad c_j\ge 0,$0

such that

$b_i\ge 0,\qquad c_j\ge 0,$1

with $b_i\ge 0,\qquad c_j\ge 0,$2 in the mammillary form above. Because $b_i\ge 0,\qquad c_j\ge 0,$3, the state basis is essentially fixed by the physiology and topology; the problem is therefore not whether $b_i\ge 0,\qquad c_j\ge 0,$4 has some realization of dimension $b_i\ge 0,\qquad c_j\ge 0,$5, but whether it has one with exactly this mammillary pattern (Beltrami et al., 14 Jul 2025).

2. Transfer-function structure and exact realizability theorem

For the general $b_i\ge 0,\qquad c_j\ge 0,$6-compartment mammillary model, the transfer function is

$b_i\ge 0,\qquad c_j\ge 0,$7

where

$b_i\ge 0,\qquad c_j\ge 0,$8

Thus the numerator is automatically monic of degree $b_i\ge 0,\qquad c_j\ge 0,$9, so any such realization has relative degree $a_{ij}\ge 0\quad \text{for } i\neq j,$0 (Beltrami et al., 14 Jul 2025).

The central realizability theorem states that there exists a unique parameter vector $a_{ij}\ge 0\quad \text{for } i\neq j,$1 such that

$a_{ij}\ge 0\quad \text{for } i\neq j,$2

if and only if two conditions hold:

1. $a_{ij}\ge 0\quad \text{for } i\neq j,$3 has relative degree $a_{ij}\ge 0\quad \text{for } i\neq j,$4;
2. the numerator $a_{ij}\ge 0\quad \text{for } i\neq j,$5 is a monic polynomial with simple, real, and nonzero roots.

Equivalently, if

$a_{ij}\ge 0\quad \text{for } i\neq j,$6

with distinct real nonzero roots $a_{ij}\ge 0\quad \text{for } i\neq j,$7, then $a_{ij}\ge 0\quad \text{for } i\neq j,$8 admits a unique mammillary realization of the prescribed form (Beltrami et al., 14 Jul 2025).

This characterization is unusually rigid. The numerator alone fixes the peripheral return rates: $a_{ij}\ge 0\quad \text{for } i\neq j,$9 The remaining parameters are then recovered by evaluating the denominator at special points. Writing

$a_{ii}+\sum_{j\neq i} a_{ji}\le 0.$0

the constructive formulas are

$a_{ii}+\sum_{j\neq i} a_{ji}\le 0.$1

Uniqueness follows because each parameter is explicitly determined by $a_{ii}+\sum_{j\neq i} a_{ji}\le 0.$2 (Beltrami et al., 14 Jul 2025).

The proof is elementary and constructive. Necessity comes from the explicit numerator

$a_{ii}+\sum_{j\neq i} a_{ji}\le 0.$3

whose zeros are $a_{ii}+\sum_{j\neq i} a_{ji}\le 0.$4, hence real, nonzero, and simple when the $a_{ii}+\sum_{j\neq i} a_{ji}\le 0.$5 are distinct. Sufficiency is obtained by matching the numerator through $a_{ii}+\sum_{j\neq i} a_{ji}\le 0.$6, then observing that both $a_{ii}+\sum_{j\neq i} a_{ji}\le 0.$7 and $a_{ii}+\sum_{j\neq i} a_{ji}\le 0.$8 are monic degree-$a_{ii}+\sum_{j\neq i} a_{ji}\le 0.$9 polynomials and therefore coincide if they agree at $$A= \begin{bmatrix} -k_{10}-\sum_{i=2}^n k_{1i} & k_{21} & k_{31} & \cdots & k_{n1}\ k_{12} & -k_{21} & 0 & \cdots & 0\ k_{13} & 0 & -k_{31} & \cdots & 0\ \vdots & \vdots & \vdots & \ddots & \vdots\ k_{1n} & 0 & 0 & \cdots & -k_{n1} \end{bmatrix}, \qquad B=e_1=[1,0,\dots,0]^T,\qquad C=e_1^T=[1,0,\dots,0].$$0 distinct points, chosen as

$$A= \begin{bmatrix} -k_{10}-\sum_{i=2}^n k_{1i} & k_{21} & k_{31} & \cdots & k_{n1}\ k_{12} & -k_{21} & 0 & \cdots & 0\ k_{13} & 0 & -k_{31} & \cdots & 0\ \vdots & \vdots & \vdots & \ddots & \vdots\ k_{1n} & 0 & 0 & \cdots & -k_{n1} \end{bmatrix}, \qquad B=e_1=[1,0,\dots,0]^T,\qquad C=e_1^T=[1,0,\dots,0].$$1

The identities

$$A= \begin{bmatrix} -k_{10}-\sum_{i=2}^n k_{1i} & k_{21} & k_{31} & \cdots & k_{n1}\ k_{12} & -k_{21} & 0 & \cdots & 0\ k_{13} & 0 & -k_{31} & \cdots & 0\ \vdots & \vdots & \vdots & \ddots & \vdots\ k_{1n} & 0 & 0 & \cdots & -k_{n1} \end{bmatrix}, \qquad B=e_1=[1,0,\dots,0]^T,\qquad C=e_1^T=[1,0,\dots,0].$$2

and

$$A= \begin{bmatrix} -k_{10}-\sum_{i=2}^n k_{1i} & k_{21} & k_{31} & \cdots & k_{n1}\ k_{12} & -k_{21} & 0 & \cdots & 0\ k_{13} & 0 & -k_{31} & \cdots & 0\ \vdots & \vdots & \vdots & \ddots & \vdots\ k_{1n} & 0 & 0 & \cdots & -k_{n1} \end{bmatrix}, \qquad B=e_1=[1,0,\dots,0]^T,\qquad C=e_1^T=[1,0,\dots,0].$$3

yield the recovery formulas directly (Beltrami et al., 14 Jul 2025).

3. Positive mammillary realizability and parameter synthesis

A positive mammillary realization requires strict positivity: $$A= \begin{bmatrix} -k_{10}-\sum_{i=2}^n k_{1i} & k_{21} & k_{31} & \cdots & k_{n1}\ k_{12} & -k_{21} & 0 & \cdots & 0\ k_{13} & 0 & -k_{31} & \cdots & 0\ \vdots & \vdots & \vdots & \ddots & \vdots\ k_{1n} & 0 & 0 & \cdots & -k_{n1} \end{bmatrix}, \qquad B=e_1=[1,0,\dots,0]^T,\qquad C=e_1^T=[1,0,\dots,0].$$4 together with the ordering

$$A= \begin{bmatrix} -k_{10}-\sum_{i=2}^n k_{1i} & k_{21} & k_{31} & \cdots & k_{n1}\ k_{12} & -k_{21} & 0 & \cdots & 0\ k_{13} & 0 & -k_{31} & \cdots & 0\ \vdots & \vdots & \vdots & \ddots & \vdots\ k_{1n} & 0 & 0 & \cdots & -k_{n1} \end{bmatrix}, \qquad B=e_1=[1,0,\dots,0]^T,\qquad C=e_1^T=[1,0,\dots,0].$$5

which is taken without loss of generality because one may permute the peripheral states (Beltrami et al., 14 Jul 2025).

The positive realizability theorem sharpens the unrestricted theorem. There exists a unique, strictly positive $$A= \begin{bmatrix} -k_{10}-\sum_{i=2}^n k_{1i} & k_{21} & k_{31} & \cdots & k_{n1}\ k_{12} & -k_{21} & 0 & \cdots & 0\ k_{13} & 0 & -k_{31} & \cdots & 0\ \vdots & \vdots & \vdots & \ddots & \vdots\ k_{1n} & 0 & 0 & \cdots & -k_{n1} \end{bmatrix}, \qquad B=e_1=[1,0,\dots,0]^T,\qquad C=e_1^T=[1,0,\dots,0].$$6 such that

$$A= \begin{bmatrix} -k_{10}-\sum_{i=2}^n k_{1i} & k_{21} & k_{31} & \cdots & k_{n1}\ k_{12} & -k_{21} & 0 & \cdots & 0\ k_{13} & 0 & -k_{31} & \cdots & 0\ \vdots & \vdots & \vdots & \ddots & \vdots\ k_{1n} & 0 & 0 & \cdots & -k_{n1} \end{bmatrix}, \qquad B=e_1=[1,0,\dots,0]^T,\qquad C=e_1^T=[1,0,\dots,0].$$7

if and only if:

1. $$A= \begin{bmatrix} -k_{10}-\sum_{i=2}^n k_{1i} & k_{21} & k_{31} & \cdots & k_{n1}\ k_{12} & -k_{21} & 0 & \cdots & 0\ k_{13} & 0 & -k_{31} & \cdots & 0\ \vdots & \vdots & \vdots & \ddots & \vdots\ k_{1n} & 0 & 0 & \cdots & -k_{n1} \end{bmatrix}, \qquad B=e_1=[1,0,\dots,0]^T,\qquad C=e_1^T=[1,0,\dots,0].$$8 has relative degree $$A= \begin{bmatrix} -k_{10}-\sum_{i=2}^n k_{1i} & k_{21} & k_{31} & \cdots & k_{n1}\ k_{12} & -k_{21} & 0 & \cdots & 0\ k_{13} & 0 & -k_{31} & \cdots & 0\ \vdots & \vdots & \vdots & \ddots & \vdots\ k_{1n} & 0 & 0 & \cdots & -k_{n1} \end{bmatrix}, \qquad B=e_1=[1,0,\dots,0]^T,\qquad C=e_1^T=[1,0,\dots,0].$$9;
2. the numerator of $1$0 is monic with simple, real, and strictly negative roots;
3. $1$1;
4. for every root $1$2 of the numerator,

$1$3

Since

$1$4

conditions 2–4 are exactly the sign conditions needed to make all parameters strictly positive (Beltrami et al., 14 Jul 2025).

The paper also gives an explicit synthesis algorithm. For a transfer function $1$5 of order $1$6, one verifies relative degree $1$7, factors $1$8, checks that $1$9 is monic with $2,\dots,n$0 simple real nonzero roots, and, for strict positivity, additionally checks that all roots $2,\dots,n$1, that $2,\dots,n$2, and that $2,\dots,n$3 for all $2,\dots,n$4. Ordering the roots as

$2,\dots,n$5

one computes

$2,\dots,n$6

The output is the unique mammillary parameter vector $2,\dots,n$7, provided the existence checks pass (Beltrami et al., 14 Jul 2025).

The restrictive nature of the class is visible in its failure modes. A transfer function does not admit such a realization if the relative degree is not $2,\dots,n$8, the numerator is not monic, the numerator has repeated roots, nonreal roots, or a zero root, or, in the positive case, if the numerator roots are not all strictly negative or if $2,\dots,n$9 or some $k_{10}$0. This indicates that many stable transfer functions, including many positive or compartmental ones, are not mammillary-realizable in this sense (Beltrami et al., 14 Jul 2025).

4. Three-compartment case, spectral properties, and pharmacokinetic interpretation

The $k_{10}$1 case is developed explicitly because it is both simpler and directly relevant to pharmacokinetics. With

$k_{10}$2

the transfer function is

$k_{10}$3

with

$k_{10}$4

$k_{10}$5

If

$k_{10}$6

then

$k_{10}$7

$k_{10}$8

$k_{10}$9

$H(s)=\frac{\beta(s)}{\alpha(s)}$0

For positivity, the equivalent conditions are relative degree $H(s)=\frac{\beta(s)}{\alpha(s)}$1, a monic numerator with simple, real, strictly negative roots $H(s)=\frac{\beta(s)}{\alpha(s)}$2, and the sign pattern

$H(s)=\frac{\beta(s)}{\alpha(s)}$3

(Beltrami et al., 14 Jul 2025).

A further structural result is that positive mammillary matrices have only real, nonpositive eigenvalues. The proof uses the diagonal similarity

$H(s)=\frac{\beta(s)}{\alpha(s)}$4

with

$H(s)=\frac{\beta(s)}{\alpha(s)}$5

which yields a symmetric matrix $H(s)=\frac{\beta(s)}{\alpha(s)}$6. Since $H(s)=\frac{\beta(s)}{\alpha(s)}$7 is symmetric, its eigenvalues are real; since $H(s)=\frac{\beta(s)}{\alpha(s)}$8 is also compartmental, they lie in the closed left half-plane. Hence all poles are real and nonpositive (Beltrami et al., 14 Jul 2025).

This structure underlies the pharmacokinetic interpretation. In the mammillary form, $H(s)=\frac{\beta(s)}{\alpha(s)}$9 is the rate from central to peripheral compartment $n$0, $n$1 is the rate from peripheral $n$2 back to central, $n$3 is elimination from the body only from the central compartment, $n$4 means infusion enters the central compartment, and $n$5 means the measured concentration is in the central compartment. The result is a physiologically interpretable model of drug distribution and clearance rather than a black-box transfer-function fit (Beltrami et al., 14 Jul 2025).

5. Propofol PK/PD extension and nonuniqueness from effect-site factorization

The mammillary realization framework is extended in a propofol application by adding an effect-site compartment and obtaining a 4-state PK/PD Wiener structure. The state is

$n$6

where $n$7 are drug masses in primary, fast, and slow compartments, and $n$8 is effect-site concentration (Beltrami et al., 14 Jul 2025).

The state matrix is

$n$9

The upper-left $b_i\ge 0,\qquad c_j\ge 0,$00 block is exactly the 3-compartment mammillary PK model. The effect-site compartment is not a PK compartment in the mass-balance sense; $b_i\ge 0,\qquad c_j\ge 0,$01 does not appear as a loss term in the central compartment because this compartment models delay in effect, not elimination from the body (Beltrami et al., 14 Jul 2025).

The transfer function is

$b_i\ge 0,\qquad c_j\ge 0,$02

where

$b_i\ge 0,\qquad c_j\ge 0,$03

$b_i\ge 0,\qquad c_j\ge 0,$04

$b_i\ge 0,\qquad c_j\ge 0,$05

Thus the PK/PD transfer function factors as a mammillary PK transfer function times a first-order low-pass filter

$b_i\ge 0,\qquad c_j\ge 0,$06

The realizability theorem for this extended model requires relative degree $b_i\ge 0,\qquad c_j\ge 0,$07, a numerator with simple, real, nonzero roots, and a denominator with at least one real root; the positive version requires simple, real, negative, nonzero numerator roots, positive leading coefficient, at least one real negative denominator root $b_i\ge 0,\qquad c_j\ge 0,$08, and sign conditions on the residual cubic factor $b_i\ge 0,\qquad c_j\ge 0,$09 (Beltrami et al., 14 Jul 2025).

Unlike the pure mammillary case, the PK/PD realization is generally not unique, because any real pole of the denominator may be assigned to the effect-site filter $b_i\ge 0,\qquad c_j\ge 0,$10, with the remaining factor interpreted as the cubic PK denominator. This produces multiple positive realizations when several real negative poles satisfy the sign test. In the propofol example, different choices of $b_i\ge 0,\qquad c_j\ge 0,$11 lead either to a positive realization reproducing the Schnider parameters up to peripheral-compartment relabeling or to alternative positive realizations (Beltrami et al., 14 Jul 2025).

6. Identifiability of mammillary parameters from input-output data

A distinct but closely related line of work studies not transfer-function realizability of the canonical central-input/central-output model, but parameter identifiability for leak-free one-input/one-output mammillary models with fixed graph structure $b_i\ge 0,\qquad c_j\ge 0,$12 (Clemens et al., 27 Jun 2025). Here a mammillary model is the bidirected star

$b_i\ge 0,\qquad c_j\ge 0,$13

with compartment $b_i\ge 0,\qquad c_j\ge 0,$14 central and compartments $b_i\ge 0,\qquad c_j\ge 0,$15 peripheral. The paper distinguishes parameters that are generically globally identifiable, generically locally identifiable, and unidentifiable, and uses the term SLING for a parameter that is generically locally identifiable but not generically globally identifiable (Clemens et al., 27 Jun 2025).

For one-input/one-output, no-leak mammillary models, the five symmetry classes are

$b_i\ge 0,\qquad c_j\ge 0,$16

Their parameter-level identifiability properties are as follows (Clemens et al., 27 Jun 2025):

Family	Generically globally identifiable parameters	Other proved status
$b_i\ge 0,\qquad c_j\ge 0,$17, $b_i\ge 0,\qquad c_j\ge 0,$18	None	All parameters are SLING
$b_i\ge 0,\qquad c_j\ge 0,$19, $b_i\ge 0,\qquad c_j\ge 0,$20	$b_i\ge 0,\qquad c_j\ge 0,$21	All remaining parameters are SLING
$b_i\ge 0,\qquad c_j\ge 0,$22, $b_i\ge 0,\qquad c_j\ge 0,$23	$b_i\ge 0,\qquad c_j\ge 0,$24, $b_i\ge 0,\qquad c_j\ge 0,$25	Remaining parameters are SLING
$b_i\ge 0,\qquad c_j\ge 0,$26, $b_i\ge 0,\qquad c_j\ge 0,$27	$b_i\ge 0,\qquad c_j\ge 0,$28, $b_i\ge 0,\qquad c_j\ge 0,$29	Remaining parameters are SLING
$b_i\ge 0,\qquad c_j\ge 0,$30, $b_i\ge 0,\qquad c_j\ge 0,$31	No full classification proved	$b_i\ge 0,\qquad c_j\ge 0,$32 are SLING; other parameters are conjectured unidentifiable

The mechanism behind these results is combinatorial. For a one-input/one-output model with input $b_i\ge 0,\qquad c_j\ge 0,$33 and output $b_i\ge 0,\qquad c_j\ge 0,$34, the input-output equation is

$b_i\ge 0,\qquad c_j\ge 0,$35

or, in coefficient form,

$b_i\ge 0,\qquad c_j\ge 0,$36

For leak-free models,

$b_i\ge 0,\qquad c_j\ge 0,$37

Bortner et al. provide a combinatorial formula expressing $b_i\ge 0,\qquad c_j\ge 0,$38 and $b_i\ge 0,\qquad c_j\ge 0,$39 as sums over spanning incoming forests, and the mammillary identifiability proofs specialize this formula to the bidirected star (Clemens et al., 27 Jun 2025).

Several exact reconstruction formulas are available. If there is an edge from the unique input compartment $b_i\ge 0,\qquad c_j\ge 0,$40 to the unique output compartment $b_i\ge 0,\qquad c_j\ge 0,$41, then the corresponding parameter $b_i\ge 0,\qquad c_j\ge 0,$42 is globally identifiable, with

$b_i\ge 0,\qquad c_j\ge 0,$43

This yields $b_i\ge 0,\qquad c_j\ge 0,$44 in $b_i\ge 0,\qquad c_j\ge 0,$45 and $b_i\ge 0,\qquad c_j\ge 0,$46 in $b_i\ge 0,\qquad c_j\ge 0,$47. For $b_i\ge 0,\qquad c_j\ge 0,$48, the paper proves

$b_i\ge 0,\qquad c_j\ge 0,$49

and

$b_i\ge 0,\qquad c_j\ge 0,$50

In $b_i\ge 0,\qquad c_j\ge 0,$51, it gives the explicit formula

$b_i\ge 0,\qquad c_j\ge 0,$52

Most remaining parameters are only identifiable up to permutation of symmetric peripheral compartments, which is why SLING behavior dominates the classification (Clemens et al., 27 Jun 2025).

7. Terminological scope and adjacent realization theories

The term mammillary realization is specific to structured compartmental models and should not be conflated with other realization theories. The paper on positive Markov realizations studies a different structured subclass of positive realizations, namely companion-like Markov-form realizations for discrete-time SISO transfer functions. It does not discuss mammillary, compartmental, or star-shaped realizations explicitly, but it is relevant as an adjacent example of how restricting to a realization structure can make a minimum-dimension synthesis problem tractable while also creating a dimension gap relative to unrestricted positive realizations (Taghavian et al., 28 Feb 2025).

A second nearby but distinct literature concerns noncommutative rational functions. “Realizations of non-commutative rational functions around a matrix centre, I” develops matrix-centered noncommutative Fornasini–Marchesini realizations, proves existence and uniqueness of a minimal realization centered at an arbitrary matrix point, and studies evaluation on full matrix domains and on stably finite algebras. The paper explicitly does not use the term mammillary realization; its main framework is nc Fornasini–Marchesini, with descriptor realizations discussed only as a related secondary form (Porat et al., 2019).

A third possible source of ambiguity is geometric realization counting on the sphere. “Calligraphs and sphere realizations” studies realizations of minimally rigid graphs on

$b_i\ge 0,\qquad c_j\ge 0,$53

up to $b_i\ge 0,\qquad c_j\ge 0,$54, using moduli spaces $b_i\ge 0,\qquad c_j\ge 0,$55, calligraphic splits, and a three-integer invariant governed by a quadratic form. That work is directly relevant only if “mammillary realizations” is interpreted as sphere-based realization theory; terminologically and technically, it is not about mammillary compartmental models (Gallet et al., 2023).

Taken together, these distinctions locate mammillary realizations within structured positive and compartmental system theory. Their defining feature is not minimality up to arbitrary similarity but exact adherence to a star-shaped central/peripheral architecture with physiologically meaningful parameters. The recent literature therefore splits naturally into three questions: exact transfer-function realizability in the canonical mammillary form, parameter identifiability once a mammillary graph and input-output placement are fixed, and comparison with adjacent structured realization classes whose algebraic and optimization tools may be methodologically informative but are not themselves mammillary (Beltrami et al., 14 Jul 2025, Clemens et al., 27 Jun 2025, Taghavian et al., 28 Feb 2025).