Deriving Exchange Time Scales For Multi-Compartment Tissues

Why Exchange Time Scales Matter

At equilibrium, compartments may have a predictable relationship between their concentrations. But that relationship does not specify how quickly the tissue approaches equilibrium. A membrane can permit the same eventual partitioning while allowing exchange to occur slowly. For a tissue with extracellular, intracellular, and endosomal spaces, multiple exchange steps can also produce more than one characteristic time scale.

The aim here is to derive those time scales for a passive, linear exchange system. The central calculation is an eigenvalue calculation: each positive decay rate describes a mode of redistribution, and its reciprocal is that mode’s relaxation time. These times describe exchange within the tissue model, not whole-body drug half-lives.

Set Up the Exchange System

Index the extracellular, intracellular, and endosomal compartments by \(e\), \(i\), and \(n\), respectively. Let \(C_j\) be the concentration in compartment \(j\), and let \(K_j>0\) be its equilibrium partition factor relative to a common reference. Define the transformed concentration \(y_j=C_j/K_j\). At equilibrium, the transformed concentrations are equal, while the actual concentrations may differ according to the \(K_j\) values.

Let \(V_j\) be the compartment volume and define its storage capacity as \(S_j=V_jK_j\). The amount stored in compartment \(j\) is then \(V_jC_j=S_jy_j\). For exchange across each membrane, let \(G_{ei}\) and \(G_{in}\) denote positive conductances, with units of volume per time. For example, a conductance may be represented as permeability multiplied by membrane area. Under this model, the flux from \(e\) to \(i\) is \(G_{ei}(y_e-y_i)\); the flux from \(i\) to \(n\) is \(G_{in}(y_i-y_n)\).

With \(G_{ei}=a\) and \(G_{in}=b\), amount balance gives:

The same equations can be written \(\frac{d\mathbf{y}}{dt}=-D^{-1}L\mathbf{y}\), where \(D=\operatorname{diag}(S_e,S_i,S_n)\) and

The vector with all three entries equal is an equilibrium direction: if \(y_e=y_i=y_n\), both fluxes are zero. It is therefore associated with a zero eigenvalue, not a decaying mode. The other two eigenvalues of \(D^{-1}L\) are positive decay rates, denoted \(r_-\) and \(r_+\), with \(r_-\leq r_+\). The corresponding eigenvalues of the differential-equation matrix \(-D^{-1}L\) are \(-r_-\) and \(-r_+\).

Derive the Two Relaxation Rates

The decay rates are the generalized eigenvalues solving \(L\mathbf{v}=rD\mathbf{v}\). For a three-compartment chain, their sum and product are:

Thus the two nonzero rates are the roots of \(r^2-Tr+Q=0\):

These expressions follow directly from the characteristic equation. Because one eigenvalue is zero, the characteristic polynomial of \(D^{-1}L\) has the form \(r(r^2-Tr+Q)\). Its trace gives \(T\). The coefficient \(Q\), the product of the two nonzero eigenvalues, can be found by summing the three principal \(2\)-by-\(2\) minors of \(D^{-1}L\). Those minors are \(a(a+b)/(S_eS_i)\), \(ab/(S_eS_n)\), and \(b(a+b)/(S_iS_n)\) only if formed without accounting for their shared terms; collecting the correct coefficients from the characteristic polynomial yields \(Q=ab(1/(S_eS_i)+1/(S_eS_n)+1/(S_iS_n))\). Equivalently, this product follows from the nonzero generalized eigenvalues of the three-node weighted chain. Thus \(T\) has units of inverse time, \(Q\) has units of inverse time squared, and each \(r\) has units of inverse time.

The discriminant is nonnegative. In fact, \(D^{-1}L\) is similar to the symmetric matrix \(D^{-1/2}LD^{-1/2}\), whose eigenvalues are real. With positive conductances and a connected chain, \(L\) has exactly one zero direction and two positive eigenvalues in this weighted system. The two relaxation rates are therefore real and positive.

How Capacity and Permeability Shape the Modes

A larger storage capacity tends to slow concentration change for a given net flux: the same amount transferred produces a smaller change in \(y_j\). A smaller membrane conductance likewise reduces flux for a given difference in transformed concentrations. These effects can create a slow mode, but the rates depend on the full connected system. It is not generally correct to assign each membrane a separate time constant and assume those constants are the system’s eigen-times.

Interpretation, Checks, and Alternatives

For this closed system, total amount is \(M=S_ey_e+S_iy_i+S_ny_n\). Adding the three amount-balance equations cancels each internal flux, so \(dM/dt=0\). The equilibrium transformed concentration is consequently \(M/(S_e+S_i+S_n)\). Away from equilibrium, deviations can be represented as combinations of the two decaying modes and the conserved equilibrium mode. A mode’s time constant describes its exponential decay; it does not, by itself, describe a measured concentration curve or guarantee that the mode has a large observable amplitude.

Eigenvalue analysis complements, rather than replaces, direct simulation of the exchange equations. It summarizes the intrinsic linear time scales without requiring a particular initial condition. Simulation shows the actual trajectories for chosen initial concentrations, while an algebraic equilibrium-partition calculation gives the final relationship but not the approach time. A perfusion-limited representation may be appropriate when tissue exchange is effectively rapid relative to the process of interest; it cannot represent a slow membrane-exchange mode when that mode matters.

Several checks help catch errors before interpreting the results. Both calculated rates should be real and positive, and their sum and product should agree with \(T\) and \(Q\). Conductance divided by capacity must have units of inverse time. In a closed exchange calculation, total amount should remain constant. A negative rate, a complex rate for this passive chain, or a nonzero rate for the uniform equilibrium direction indicates a sign, indexing, or calculation error. If elimination or external input is added, this closed-system derivation must be modified; the zero mode and the formulas here may no longer apply.

Common Pitfall

A frequent error is to calculate \(G_{ei}/S_e\) and \(G_{in}/S_n\), call each the rate of one membrane, and treat their reciprocals as the tissue’s two relaxation times. Those ratios describe individual contributions to local concentration change, but the compartments are coupled: intracellular storage and both pathways affect the system modes. The correction is to form the coupled exchange matrix, find its nonzero eigenvalues, and take their reciprocals. Another error is to mistake the zero eigenvalue for an infinite relaxation time. It represents conserved total amount and the equilibrium direction, not a deviation that needs to decay.