Species Amounts and Concentrations in Changing-Volume Compartments

Why Volume Matters

A species can change concentration even when its amount does not change. Adding solute-free fluid dilutes a compartment; removing solute-free fluid concentrates it. Conversely, a changing amount does not by itself determine whether concentration rises or falls: the volume may be changing at the same time. A QSP model that uses concentration as its state must account for both changes.

The essential step is to write the balance for the species amount first, then convert that balance to concentration. This keeps the physical meaning of each term visible and avoids treating concentration as if it were an amount. The equations below apply to one well-mixed compartment; the same conversion can be used for each compartment in a larger model.

From an Amount Balance to a Concentration Balance

Assume that amount and volume are differentiable in time and that the compartment is well mixed, so its concentration is spatially uniform. Differentiate \(A=CV\) with respect to time. The product rule gives the general conversion:

The second term in the concentration equation is the volume-change contribution. When volume increases, it lowers concentration at fixed amount; when volume decreases, it raises concentration at fixed amount. The term is not an additional reaction. It arises from expressing an amount balance using concentration as the state.

For a compartment with fluid inflow \(Q_{\mathrm{in}}\) at concentration \(C_{\mathrm{in}}\), fluid outflow \(Q_{\mathrm{out}}\), and net biochemical production rate \(R\), a suitable amount balance is

Here \(R\) is the net amount produced by reactions per unit time, with production positive and consumption negative. The flow terms assume that the incoming fluid carries the species at concentration \(C_{\mathrm{in}}\), and that outgoing fluid carries it at the compartment concentration \(C\). Thus, the compartment is well mixed, and there is no selective removal or addition of the species beyond the listed terms. For example, \(Q_{\mathrm{in}}C_{\mathrm{in}}\) has units of amount per time when \(Q_{\mathrm{in}}\) is volume per time and concentration is amount per volume.

Substituting this amount balance into the conversion gives a concentration equation that remains valid when volume changes for any reason:

If fluid flows alone determine the volume change, with no other source or sink of volume, then \(dV/dt=Q_{\mathrm{in}}-Q_{\mathrm{out}}\). Substitution and collection of terms yields the simpler form

This simplification depends on the stated volume balance. It should not be used automatically when, for example, volume also changes through an unmodeled fluid transfer. In that case, retain the general equation and use the actual \(dV/dt\).

Worked Applications

Assumptions, Diagnostics, and Alternatives

Before using a concentration equation, check that the state represents concentration in a well-mixed compartment and that \(V(t)\) is positive. The balance must account for the processes that change species amount, while the volume equation must account for the processes that change volume. If flows alone determine volume, the equality \(dV/dt=Q_{\mathrm{in}}-Q_{\mathrm{out}}\) must be justified for that model. If volume can change independently, keep its separate rate in the concentration equation.

A useful diagnostic is to reconstruct the implied amount rate from a proposed concentration equation:

This must match the amount balance. If it does not, a volume term may be missing, duplicated, or assigned the wrong sign. Also check units: \(dC/dt\) must have units of concentration per time, while \(dA/dt\) must have units of amount per time. For positive \(C\), increasing volume with fixed amount must produce a negative concentration rate; a positive rate in that case signals an error in the conversion or its implementation.

Two state choices are common. An amount state makes conservation and species-bearing flow terms direct, but observations or rates expressed as concentrations require conversion using the current volume. A concentration state may align more naturally with concentration measurements and concentration-dependent reaction rates, but its equation must include the changing-volume contribution. Neither choice removes the need to model volume consistently. If the compartment is not well mixed, a single concentration may not represent the local concentration carried by outflow, and a spatial or multi-compartment description may be needed.

Common Mistakes

• Using only \(dC/dt=(dA/dt)/V\). This omits the change in concentration caused by changing volume. The correct conversion also subtracts \((C/V)(dV/dt)\).
• Calling the volume term a species loss. Expansion can lower concentration without removing any species amount. It changes the relationship between amount and concentration.
• Using the compartment concentration for inflow. Incoming fluid carries its own concentration \(C_{\mathrm{in}}\); the outgoing fluid carries \(C\) under the well-mixed assumption.
• Assuming volume change equals inflow minus outflow in every model. This holds only when those flows alone determine volume. Other volume sources or sinks must be represented in \(dV/dt\).
• Mixing rounded intermediate values. Use unrounded values in intermediate calculations, or ensure that displayed rounded terms produce the displayed result.