## 4 State Variables, Compartments, Directed, Graphs, Pools and Species  
We formalize and further clarify our primary nomenclature, paraphrasing classical definitions collectively in the contextual domain of biological systems, e.g., see [12; 15; 16]. A **compartment** is an amount of material \(X\) that acts kinetically (within a dynamic system) in a _homogeneously distinct_ manner. Homogeneously distinct are the keywords here. This means entities within a compartment are indistinguishable from each other, they are \"mixed completely\". If they are not, they comprise more than one compartment. A compartment is **open** if it \"leaks\" into the environment; it is **closed** if it does not (Figure 2). Most (not all) compartmental models of real systems have at least one open compartment. Every (physical) space containing compartments has a presumed volume in which the material substance (e.g., chemical \"species\", etc.) is homogeneously distributed. This **compartment volume**\(V_{i}\) might be measurably real, or it might be virtual--called an _equivalent volume_. _In other words_, in chemical, biological or other physical spaces, a compartment is an idealized store of a substance, characterized by its environment, physiochemical properties or both. All of the substance in a _given form_ qualifies as a compartment, e.g., _active_ or _inactive_ or _receptor-bound_ or _unbound_ form--one compartment for each--because they generally have different dynamics. Alternatively, all of the substance in a homogeneously distributed (well-mixed) location, or all of it in a given form and location, qualifies as compartments, e.g., the mass of unbound drug \(D_{F}\) (or hormone, lipid or ion) in blood or the mass of receptor-bound drug \(D_{R}\) in blood or either in the liver (or another organ or organelle). Similarly, the total drug mass \(D_{F}+D_{R}\) in the blood or liver (or another organ or organelle), is each a valid compartment in principle. Similarly, in modeling epidemiological systems, _Susceptible_, _Exposed_, _Infected_ and _Recovered_ (_SEIR_) populations are typically each considered valid, homogeneous compartments, as in COVID-19 modeling [17]. A **multicompartmental (MC) model** consists of a directed graph of two or more compartments interconnected, so there can be an exchange of material or change of state (into two or more compartments) among some or all of them and possible **leaks** from any of them (i.e., Figure 2). **Exchange** may occur by material transgressing some physical barrier (e.g., by diffusion or membrane transport) or by undergoing some physical or chemical transformation (e.g., metabolism, inactive to active state, etc.). Each **compartment** is represented by a single **state variable** in a multicompartmental (MC) model. Changes in the location of material generate different compartments (different state variables) for the same material, typically with different volumes. Changes in material form (e.g., precursor to product, carbohydrate to metabolite, infected to recovered, etc.) generate different compartments (different state variables)--possibly in the same \"space\" or \"container\" and with the same volume, and so on. Figure 2: Two-compartment model representation (linear or nonlinear) illustrating one OPEN compartment, with a LEAK and a CLOSED compartment (no LEAK) exchanging material (solid arrows) with each other and possibly other compartments not shown (dotted arrows). **Circles** are commonly used to represent compartments **arrows** to represent unidirectional transfers--which include inputs, influxes and effluxes (leaks)--and **dashed lines**, with a small open-circle endpoint, denote measured outputs (Figure 3). **Directed graphs (digraphs)** also are used to represent MC models, with **nodes** (vertices) representing compartments and **edges** as the arrows interconnecting compartments (nodes) or the environment, as in Figure 3. For linear systems, arrows and edges are typically represented by constant parameters (e.g., fractional transfer rates \(k_{ij}\)).