State Variables for Luminal Drug in Segmental Models

Why State Definitions Matter

A segmented gut model describes drug in several luminal regions. Before specifying how drug moves or changes form, the model needs a precise account of what is present in each segment. A state variable should identify both the drug form being represented and whether the quantity is an amount or a concentration-like measure. These distinctions determine how reported measurements and physiological volumes can be used without confusing a change in concentration with a change in drug amount.

The earlier material on the architecture of segmented gut absorption models establishes the use of indexed regions as model compartments. Here, let \(j\) identify one such segment, and let \(t\) denote time. The definitions below apply separately to every segment. They do not specify any particular transit, dissolution, or absorption process.

For example, amounts might be recorded in milligrams (mg), volumes in millilitres (mL), and dissolved concentrations in mg/mL. If amounts are in micromoles and volume is in litres, concentrations are in micromoles per litre. The numerator and denominator units must be stated and compatible. If a formulation amount is expressed as a salt mass while the model tracks active-moiety mass, those quantities must first be converted to a common drug basis; they cannot be added as though they were identical.

The distinction between \(C_j\) and \(Q_j\) is important. Dissolved drug is distributed in the fluid, so \(D_j/V_j\) has the interpretation of dissolved mass per unit fluid volume, given a well-mixed representation of that fluid. Undissolved drug may consist of particles or other material that does not occupy the modeled fluid uniformly. Dividing its amount by \(V_j\) gives a useful segment-level loading, but does not establish a physical concentration inside the particles. If a model separately defines a solid-phase volume, a solid-phase concentration could be defined using that volume; it is a different quantity and requires that additional volume definition.

Amounts and Volume-Based Representations

Amounts are the direct bookkeeping quantities for how much drug is assigned to a state. They do not change merely because the reference fluid volume changes. Concentration-based representations are often convenient when comparing to measurements or expressing a spatially averaged dissolved level, but they require a volume to recover the amount. For each segment,

The concentration conversion is only valid for the volume that actually corresponds to the state. A segment’s available luminal water is not necessarily the same as an anatomical volume, an administered fluid volume, or a sample volume. The physiological material on regional luminal fluid volumes and available luminal water provides the relevant context for choosing \(V_j\); the model must still state which estimate it uses and what that volume represents.

The sum \(D_j+S_j\) is the total amount assigned to the two luminal drug forms in segment \(j\), provided both amounts use the same drug basis. Dividing that sum by \(V_j\) yields a total luminal loading per unit fluid volume. It is not the dissolved concentration unless \(S_j=0\). Treating undissolved drug as if it were dissolved would imply that it contributes to the fluid concentration in the same way, which is not what the definitions mean.

Concentration States Require an Explicit Volume

A model can use \(C_j\) rather than \(D_j\) as its dissolved-drug state, or \(Q_j\) rather than \(S_j\) as its undissolved-loading state. Either choice can represent the same information if the volume is known and positive. When the volume is time-varying, it must be available at the same time as the concentration to determine the amount. A concentration alone cannot identify the amount in a segment of unknown volume.

The relation between dissolved amount and concentration also explains why the two variables can behave differently as the fluid volume changes. For differentiable \(D_j(t)\) and positive, differentiable \(V_j(t)\), apply the product rule to \(D_j=C_jV_j\):

Rearranging gives

This identity is a consequence of the definition \(C_j=D_j/V_j\), not a particular drug-process model. It shows that concentration can change because the amount changes, because the volume changes, or because both change. If the amount is momentarily constant and volume increases, concentration decreases. A concentration-based model that treats volume as fixed when it is not will therefore imply an incorrect amount.

Volume Changes and State Interpretation

For a fixed amount, changing the reference volume changes its volume-based representation. Consider an amount of dissolved drug that remains at 4.00 mg while the available fluid volume changes from 40.0 mL to 80.0 mL. Initially, \(C_j=4.00/40.0=0.100\text{ mg/mL}\). At the larger volume, \(C_j=4.00/80.0=0.0500\text{ mg/mL}\). The concentration halves even though no drug amount was removed.

Conversely, if a model specifies \(C_j=0.100\text{ mg/mL}\) at both volumes, it represents 4.00 mg at 40.0 mL and 8.00 mg at 80.0 mL. Holding concentration fixed is therefore not equivalent to holding amount fixed. This is a modeling choice with a physical implication, not a harmless change in notation.

Choosing and Auditing a State Representation

Amount-based states and concentration-based states are alternative representations, not different drug forms. Amount states make the quantity of drug explicit and avoid requiring a volume conversion merely to interpret how much drug is present. Concentration states can be useful when the model or available observations are naturally expressed per volume. They require careful tracking of the volume used in the conversion, especially if that volume varies by segment or time.

A practical audit begins by writing the units next to every state and volume. Multiplying a concentration in mg/mL by a volume in mL must yield an amount in mg. If it does not, a conversion factor or a unit definition is missing. Next, check that each segment’s volume refers to the same region and fluid pool as its concentration. A concentration measured in a sample cannot automatically be multiplied by an unrelated whole-segment volume.

Also check whether the model is assuming a single well-mixed dissolved concentration within each segment. This is a compartment-level approximation. If concentrations vary substantially within a segment, one average \(C_j\) may not represent local conditions adequately. A further limitation is that the definitions alone do not identify the amount that is actually in solution at saturation, the particle properties, or how drug changes between forms. Those require additional model structure and information.

• Missing or zero volume: \(C_j=D_j/V_j\) and \(Q_j=S_j/V_j\) require \(V_j>0\). If a segment has no defined fluid volume, a volume-based state is undefined; an amount state may still be meaningful.
• Unit mismatch: Do not multiply mg/mL by a volume in litres without converting units, or add amounts expressed on incompatible salt and active-moiety bases.
• Mislabelled loading: Calling \(S_j/V_j\) a dissolved concentration wrongly attributes undissolved drug to the aqueous phase.
• Inconsistent time updates: If \(V_j(t)\) changes, recalculate the amount or concentration from the state actually held fixed; do not silently keep both fixed.
• Unclear volume meaning: An anatomical segment volume and its available luminal fluid volume are not interchangeable without a stated modeling rationale.

A reliable specification for each segment therefore identifies the drug form, whether the state is an amount or volume-based quantity, the units, the reference volume, and whether that volume varies with time. These choices make the representation auditable before any process equations are added.