--- title: Mammillary Realizations in Compartmental Systems url: https://www.emergentmind.com/topics/mammillary-realizations type: topic --- # Mammillary Realizations in Compartmental Systems Searching arXiv for recent 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 [2507.10138]. 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 [2507.10138] [2506.21889]. ## 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 [2507.10138]. 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 \(k_{10}\) [2507.10138]. 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 \[ p=(k_{10},k_{21},\dots,k_{n1},k_{12},\dots,k_{1n})\in \mathbb R^{2n-1} \] such that \[ H(s)=C(sI-A)^{-1}B \] with \(A,B,C\) in the mammillary form above. Because \(B=C^T=e_1\), the state basis is essentially fixed by the physiology and topology; the problem is therefore not whether \(H\) has some realization of dimension \(n\), but whether it has one with exactly this mammillary pattern [2507.10138]. ## 2. Transfer-function structure and exact realizability theorem For the general \(n\)-compartment mammillary model, the transfer function is \[ T_p(s)=\frac{(s+k_{21})(s+k_{31})\cdots(s+k_{n1})}{\chi_p(s)}, \] where \[ \chi_p(s)=\det(sI-A_p). \] Thus the numerator is automatically monic of degree \(n-1\), so any such realization has relative degree \(1\) [2507.10138]. The central realizability theorem states that there exists a unique parameter vector \(p\) such that \[ T_p(s)=H(s) \] if and only if two conditions hold: 1. \(H\) has relative degree \(1\); 2. the numerator \(\beta(s)\) is a monic polynomial with simple, real, and nonzero roots. Equivalently, if \[ \beta(s)=(s-z_2)(s-z_3)\cdots(s-z_n), \] with distinct real nonzero roots \(z_2>\cdots>z_n\), then \(H\) admits a unique mammillary realization of the prescribed form [2507.10138]. This characterization is unusually rigid. The numerator alone fixes the peripheral return rates: \[ k_{i1}=-z_i,\qquad i=2,\dots,n. \] The remaining parameters are then recovered by evaluating the denominator at special points. Writing \[ \beta_i(s):=z_i\,\widehat{\beta(s)}^{\,i},\qquad \widehat{\beta(s)}^{\,i}:=\frac{\beta(s)}{s-z_i}, \] the constructive formulas are \[ k_{10}=\frac{\alpha(0)}{\beta(0)},\qquad k_{1i}=\frac{\alpha(z_i)}{\beta_i(z_i)},\qquad i=2,\dots,n. \] Uniqueness follows because each parameter is explicitly determined by \(H\) [2507.10138]. The proof is elementary and constructive. Necessity comes from the explicit numerator \[ (s+k_{21})(s+k_{31})\cdots(s+k_{n1}), \] whose zeros are \(-k_{i1}\), hence real, nonzero, and simple when the \(k_{i1}\) are distinct. Sufficiency is obtained by matching the numerator through \(k_{i1}=-z_i\), then observing that both \(\alpha\) and \(\chi_p\) are monic degree-\(n\) polynomials and therefore coincide if they agree at \(n\) distinct points, chosen as \[ s=0,\quad s=z_2,\dots,z_n. \] The identities \[ \chi_p(0)=n_p(0)k_{10} \] and \[ \chi_p(-k_{i1})=k_{1i}\,n_{p,i}(s) \] yield the recovery formulas directly [2507.10138]. ## 3. Positive mammillary realizability and parameter synthesis A positive mammillary realization requires strict positivity: \[ k_{10}>0,\qquad k_{ij}>0 \text{ for all relevant } i\neq j, \] together with the ordering \[ k_{21}0\); 4. for every root \(z_i\) of the numerator, \[ \frac{\beta_i(z_i)}{\alpha(z_i)}>0. \] Since \[ k_{10}=\frac{\alpha(0)}{\beta(0)},\qquad k_{i1}=-z_i,\qquad k_{1i}=\frac{\alpha(z_i)}{\beta_i(z_i)}, \] conditions 2–4 are exactly the sign conditions needed to make all parameters strictly positive [2507.10138]. The paper also gives an explicit synthesis algorithm. For a transfer function \(H(s)=\beta(s)/\alpha(s)\) of order \(n\), one verifies relative degree \(=1\), factors \(\beta(s)\), checks that \(\beta\) is monic with \(n-1\) simple real nonzero roots, and, for strict positivity, additionally checks that all roots \(z_i<0\), that \(H(0)>0\), and that \(\beta_i(z_i)/\alpha(z_i)>0\) for all \(i\). Ordering the roots as \[ z_2>z_3>\cdots>z_n, \] one computes \[ k_{10}=\frac{\alpha(0)}{\beta(0)},\qquad k_{i1}=-z_i,\qquad k_{1i}=\frac{\alpha(z_i)}{\beta_i(z_i)}. \] The output is the unique mammillary parameter vector \(p\), provided the existence checks pass [2507.10138]. 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 \(1\), 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 \(H(0)\le 0\) or some \(\beta_i(z_i)/\alpha(z_i)\le 0\). This indicates that many stable transfer functions, including many positive or compartmental ones, are not mammillary-realizable in this sense [2507.10138]. ## 4. Three-compartment case, spectral properties, and pharmacokinetic interpretation The \(3\times 3\) case is developed explicitly because it is both simpler and directly relevant to pharmacokinetics. With \[ A= \begin{bmatrix} -k_{10}-k_{12}-k_{13} & k_{21} & k_{31}\\ k_{12} & -k_{21} & 0\\ k_{13} & 0 & -k_{31} \end{bmatrix}, \qquad B=\begin{bmatrix}1\\0\\0\end{bmatrix}, \qquad C=\begin{bmatrix}1&0&0\end{bmatrix}, \] the transfer function is \[ C(sI-A)^{-1}B=\frac{(k_{21}+s)(k_{31}+s)}{\chi(s)}, \] with \[ \chi(s)=s^3+(k_{10}+k_{12}+k_{13}+k_{21}+k_{31})s^2 \] \[ \quad +(k_{10}k_{21}+k_{13}k_{21}+k_{10}k_{31}+k_{12}k_{31}+k_{21}k_{31})s +k_{10}k_{31}k_{21}. \] If \[ H(s)=\frac{\beta(s)}{\alpha(s)},\qquad \beta(s)=(s-z_2)(s-z_3),\quad z_2>z_3, \] then \[ k_{21}=-z_2,\qquad k_{31}=-z_3, \] \[ k_{10}=\frac{\alpha(0)}{z_2 z_3}, \] \[ k_{12}=\frac{\alpha(z_2)}{z_2(z_2-z_3)}, \] \[ k_{13}=-\frac{\alpha(z_3)}{z_3(z_2-z_3)}. \] For positivity, the equivalent conditions are relative degree \(1\), a monic numerator with simple, real, strictly negative roots \(z_2>z_3\), and the sign pattern \[ \alpha(0)>0,\qquad \alpha(z_2)<0,\qquad \alpha(z_3)>0 \] [2507.10138]. A further structural result is that positive mammillary matrices have only real, nonpositive eigenvalues. The proof uses the diagonal similarity \[ DAD^{-1}=\tilde A \] with \[ D=\operatorname{diag}\!\left(1,\sqrt{\frac{k_{21}}{k_{12}}},\sqrt{\frac{k_{31}}{k_{13}}},\dots,\sqrt{\frac{k_{n1}}{k_{1n}}}\right), \] which yields a symmetric matrix \(\tilde A\). Since \(\tilde A\) is symmetric, its eigenvalues are real; since \(A\) is also compartmental, they lie in the closed left half-plane. Hence all poles are real and nonpositive [2507.10138]. This structure underlies the pharmacokinetic interpretation. In the mammillary form, \(k_{1i}\) is the rate from central to peripheral compartment \(i\), \(k_{i1}\) is the rate from peripheral \(i\) back to central, \(k_{10}\) is elimination from the body only from the central compartment, \(B=e_1\) means infusion enters the central compartment, and \(C=e_1^T\) 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 [2507.10138]. ## 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 \[ x(t)=\begin{bmatrix}q_1(t)&q_2(t)&q_3(t)&C_e(t)\end{bmatrix}^T, \] where \(q_1,q_2,q_3\) are drug masses in primary, fast, and slow compartments, and \(C_e\) is effect-site concentration [2507.10138]. The state matrix is \[ A= \begin{bmatrix} -k_{10}-k_{12}-k_{13} & k_{21} & k_{31} & 0\\ k_{12} & -k_{21} & 0 & 0\\ k_{13} & 0 & -k_{31} & 0\\ k_{1e}/V_1 & 0 & 0 & -k_{e0} \end{bmatrix}, \quad B=\begin{bmatrix}1\\0\\0\\0\end{bmatrix}, \quad C=\begin{bmatrix}0&0&0&1\end{bmatrix}. \] The upper-left \(3\times 3\) block is exactly the 3-compartment mammillary PK model. The effect-site compartment is not a PK compartment in the mass-balance sense; \(k_{1e}\) does not appear as a loss term in the central compartment because this compartment models delay in effect, not elimination from the body [2507.10138]. The transfer function is \[ C(sI-A)^{-1}B = \frac{\frac{k_{1e}}{V_1}(s+k_{21})(s+k_{31})}{(s+k_{e0})(s^3+\mu_2s^2+\mu_1s+\mu_0)}, \] where \[ \mu_2=k_{10}+k_{12}+k_{13}+k_{21}+k_{31}, \] \[ \mu_1=k_{10}k_{21}+k_{13}k_{21}+k_{10}k_{31}+k_{12}k_{31}+k_{21}k_{31}, \] \[ \mu_0=k_{10}k_{21}k_{31}. \] Thus the PK/PD transfer function factors as a mammillary PK transfer function times a first-order low-pass filter \[ \frac{k_{1e}/V_1}{s+k_{e0}}. \] The realizability theorem for this extended model requires relative degree \(2\), 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 \(z_0\), and sign conditions on the residual cubic factor \(a(s)\) [2507.10138]. 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 \(s+k_{e0}\), 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 \(z_0\) lead either to a positive realization reproducing the Schnider parameters up to peripheral-compartment relabeling or to alternative positive realizations [2507.10138]. ## 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 \(\mathcal M_n(i,j)\) [2506.21889]. Here a mammillary model is the bidirected star \[ V=\{1,2,\dots,n\},\qquad E=\{1\leftrightarrows 2,\;1\leftrightarrows 3,\;\dots,\;1\leftrightarrows n\}, \] with compartment \(1\) central and compartments \(2,\dots,n\) 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 [2506.21889]. For one-input/one-output, no-leak mammillary models, the five symmetry classes are \[ \mathcal M_n(1,1),\quad \mathcal M_n(1,2),\quad \mathcal M_n(2,1),\quad \mathcal M_n(2,2),\quad \mathcal M_n(2,3). \] Their parameter-level identifiability properties are as follows [2506.21889]: | Family | Generically globally identifiable parameters | Other proved status | |---|---|---| | \(\mathcal M_n(1,1)\), \(n\ge 3\) | None | All parameters are SLING | | \(\mathcal M_n(1,2)\), \(n\ge 4\) | \(k_{21}\) | All remaining parameters are SLING | | \(\mathcal M_n(2,1)\), \(n\ge 4\) | \(k_{12}\), \(k_{21}\) | Remaining parameters are SLING | | \(\mathcal M_n(2,2)\), \(n\ge 4\) | \(k_{12}\), \(k_{21}\) | Remaining parameters are SLING | | \(\mathcal M_n(2,3)\), \(n\ge 5\) | No full classification proved | \(k_{14},\dots,k_{1n}\) are SLING; other parameters are conjectured unidentifiable | The mechanism behind these results is combinatorial. For a one-input/one-output model with input \(j\) and output \(i\), the input-output equation is \[ \det(\partial I-A)\,y_i = \sum_{j\in In}(-1)^{i+j}\det\bigl[(\partial I-A)^{j,i}\bigr]u_j, \] or, in coefficient form, \[ y_i^{(n)} + c_{n-1} y_i^{(n-1)} + \dots + c_1 y_i' + c_0 y_i = d_{n-1}u_j^{(n-1)} + \dots + d_1u_j' + d_0u_j. \] For leak-free models, \[ c_0=0,\qquad d_{n-1}=0. \] Bortner et al. provide a combinatorial formula expressing \(c_k\) and \(d_k\) as sums over spanning incoming forests, and the mammillary identifiability proofs specialize this formula to the bidirected star [2506.21889]. Several exact reconstruction formulas are available. If there is an edge from the unique input compartment \(i\) to the unique output compartment \(j\), then the corresponding parameter \(k_{ji}\) is globally identifiable, with \[ d_{n-2}=k_{ji}. \] This yields \(k_{21}=d_{n-2}\) in \(\mathcal M_n(1,2)\) and \(k_{12}=d_{n-2}\) in \(\mathcal M_n(2,1)\). For \(\mathcal M_n(2,2)\), the paper proves \[ k_{12}=c_{n-1}-d_{n-2}, \] and \[ k_{21}=d_{n-2}-\frac{c_{n-2}-d_{n-3}}{c_{n-1}-d_{n-2}}. \] In \(\mathcal M_4(2,3)\), it gives the explicit formula \[ k_{14}=\frac{d_0}{d_1}. \] Most remaining parameters are only identifiable up to permutation of symmetric peripheral compartments, which is why SLING behavior dominates the classification [2506.21889]. ## 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 [2502.21102]. 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 [1905.11304]. A third possible source of ambiguity is geometric realization counting on the sphere. “Calligraphs and sphere realizations” studies realizations of minimally rigid graphs on \[ S^2_{\mathbb C}=\{(x,y,z)\in \mathbb C^3 \mid x^2+y^2+z^2=1\} \] up to \(\mathrm{SO}_3(\mathbb C)\), using moduli spaces \(\overline M_{0,2n}\), 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 [2308.15305]. 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 [2507.10138] [2506.21889] [2502.21102]. Source: https://www.emergentmind.com/topics/mammillary-realizations