Why Binding Changes Tissue Storage
A perfusion-limited tissue is often described by a relationship between blood concentration and tissue concentration. Reversible binding adds an important distinction: drug that is bound in tissue contributes to the amount stored there, but it does not necessarily contribute to the free concentration that drives exchange with blood. Thus, greater binding can increase the tissue’s total drug concentration without changing the free tissue concentration that is in equilibrium with a specified blood concentration. It can also slow the change in tissue concentration during distribution and washout.
The results below use a single perfused tissue and assume that free drug equilibrates rapidly between tissue and its local blood. Concentrations are expressed on a consistent blood reference basis; for simplicity, explicit blood binding is excluded. Binding refers to reversible drug binding within tissue, and the linear model assumes that the bound amount is proportional to free tissue concentration. These assumptions let us isolate the effect of tissue binding on partitioning and equilibration.
The coefficients have different meanings. \(K_{T:b,u}\) describes the equilibrium ratio of free tissue concentration to the specified blood concentration. \(R_T\) describes how much total tissue drug is present per unit free tissue concentration. Their product, \(K_{T:b,\mathrm{eff}}\), describes total tissue concentration relative to blood concentration under the stated assumptions. It is not generally correct to use the effective coefficient as though it were a free-concentration ratio.
From Binding to a Perfusion-Limited Time Course
For a tissue of volume \(V_T\), the amount of drug in tissue is \(A_T=V_TC_{T,\mathrm{tot}}\). Under linear binding and rapid local equilibrium, \(A_T=V_TR_TC_{T,u}\). Let \(Q\) be tissue blood flow and \(C_A\) the arterial concentration. The perfusion-limited model takes venous concentration to equal the blood concentration in equilibrium with the tissue free concentration: \(C_V=C_{T,u}/K_{T:b,u}\). The tissue mass balance is \(dA_T/dt=Q(C_A-C_V)\). Substituting the binding relationship gives the tissue equation below.
This equation assumes constant \(V_T\), \(Q\), and linear binding; no metabolism or other tissue loss; rapid reversible binding relative to perfusion-driven change; and perfusion-limited exchange. For a constant arterial concentration, the equilibrium condition is \(dC_{T,u}/dt=0\). It follows that \(C_{T,u,\mathrm{ss}}=K_{T:b,u}C_A\), while total tissue concentration is \(C_{T,\mathrm{tot,ss}}=R_TK_{T:b,u}C_A\). Therefore, when arterial concentration is externally held fixed, binding increases the equilibrium total tissue concentration, but it does not change the equilibrium free tissue concentration under this linear model.
The same equation also yields the equilibration time. Rearranging for constant \(C_A\) gives a first-order approach to equilibrium. If the tissue starts at \(C_{T,u}(0)\), the difference from equilibrium decays exponentially. The time constant \(\tau\) is the time required for that difference to fall to \(1/e\) of its initial value; the half-time is \(\ln(2)\tau\).
This derivation shows why binding slows equilibration: at a given free tissue concentration, greater \(R_T\) means more total drug must enter or leave the tissue to produce the same change in free concentration. The tissue’s equilibrium free concentration is set by the blood concentration and \(K_{T:b,u}\), but its storage capacity and time constant increase with \(R_T\). The equation assumes that the arterial concentration is constant for the time course being considered; if it varies, the tissue responds to a changing input rather than approaching one fixed equilibrium.
Worked Example: Calculating Binding-Adjusted Partitioning
Hypothetical data. A tissue has \(K_{T:b,u}=0.80\) and \(K_{\mathrm{bind}}=4.0\). The blood concentration is held at \(2.0\ \mathrm{mg/L}\). Calculate the tissue free and total concentrations and the effective partition coefficient.
First calculate the binding capacity factor: \(R_T=1+K_{\mathrm{bind}}=1+4.0=5.0\). The tissue free concentration is \(C_{T,u}=K_{T:b,u}C_B=0.80(2.0)=1.6\ \mathrm{mg/L}\). Total concentration is \(C_{T,\mathrm{tot}}=R_TC_{T,u}=5.0(1.6)=8.0\ \mathrm{mg/L}\). Equivalently, the effective coefficient is \(K_{T:b,\mathrm{eff}}=R_TK_{T:b,u}=5.0(0.80)=4.0\), and \(K_{T:b,\mathrm{eff}}C_B=4.0(2.0)=8.0\ \mathrm{mg/L}\).
Thus, under the stated fixed-blood condition, binding raises total tissue concentration to \(8.0\ \mathrm{mg/L}\), while free tissue concentration remains \(1.6\ \mathrm{mg/L}\). With no binding, \(R_T=1\), so total tissue concentration would also be \(1.6\ \mathrm{mg/L}\).
Equilibration Time and the Meaning of Equilibrium
A fixed blood concentration is not the only useful equilibrium setting. In a closed blood–tissue system, the total amount of drug is finite, so changing tissue binding can also change the common equilibrium free concentration. For example, suppose blood and tissue free concentrations are related by \(C_{T,u}=K_{T:b,u}C_B\), and blood has no explicit binding capacity in this simplified system. Conservation of drug amount gives \(A_{\mathrm{tot}}=V_BC_B+V_TR_TC_{T,u}\). Substituting the equilibrium relationship yields the equilibrium blood concentration and tissue total concentration:
Here, increasing tissue binding increases the share of the finite drug amount stored in tissue, while decreasing the equilibrium free blood concentration. The tissue total concentration does not grow without bound: as \(R_T\) becomes very large, the expression approaches \(A_{\mathrm{tot}}/V_T\). This closed-system result does not contradict the fixed-blood result. In the fixed-blood case, an external input maintains \(C_B\); in the closed system, blood concentration is allowed to fall as tissue takes up drug.
Worked Example: Comparing Equilibration Times
Hypothetical data. A tissue has volume \(V_T=12\ \mathrm{L}\), blood flow \(Q=3.0\ \mathrm{L/h}\), and \(K_{T:b,u}=0.80\). Compare equilibration half-times when \(K_{\mathrm{bind}}=4.0\) and when binding is absent. Assume constant arterial concentration and the linear, rapidly equilibrating binding model.
With binding, \(R_T=1+4.0=5.0\). The time constant is \(\tau=V_TK_{T:b,u}R_T/Q=(12\ \mathrm{L})(0.80)(5.0)/(3.0\ \mathrm{L/h})=16\ \mathrm{h}\). Therefore \(t_{1/2}=\ln(2)(16\ \mathrm{h})=11.09\ \mathrm{h}\).
Without binding, \(R_T=1\), so \(\tau=(12\ \mathrm{L})(0.80)(1)/(3.0\ \mathrm{L/h})=3.2\ \mathrm{h}\), and \(t_{1/2}=\ln(2)(3.2\ \mathrm{h})=2.218\ \mathrm{h}\). The half-time is five times larger with binding, matching the fivefold increase in \(R_T\). The equilibrium free concentration remains \(0.80C_A\) in both cases, but the bound tissue pool makes approach to that equilibrium slower.
When Binding Is Saturable
Linear binding is a useful approximation when binding sites are far from saturation over the concentrations being modeled. If binding is saturable, a common reversible equilibrium relationship is \(C_{T,\mathrm{bound}}=B_{\max}C_{T,u}/(K_d+C_{T,u})\), where \(B_{\max}\) is the maximum bound concentration and \(K_d\) is the free concentration at half of that maximum. Total tissue concentration is then free plus bound concentration. The apparent total partition coefficient depends on concentration rather than being one constant.
The second expression uses \(C_{T,u}=K_{T:b,u}C_B\). As blood concentration rises, binding sites become more occupied and the apparent total partition coefficient falls toward \(K_{T:b,u}\). The equilibrium free tissue concentration still follows the specified unbound partition relationship, but the amount of bound drug per unit free concentration is no longer constant. In this nonlinear case, the local storage capacity relevant to a small change in free concentration is the derivative \(dC_{T,\mathrm{tot}}/dC_{T,u}\), not simply a constant \(R_T\). Consequently, a single concentration-independent time constant generally does not describe the full time course.
Worked Example: Concentration-Dependent Apparent Partitioning
Hypothetical data. Let \(K_{T:b,u}=0.50\), \(B_{\max}=12\ \mathrm{mg/L}\), and \(K_d=2.0\ \mathrm{mg/L}\). Calculate the apparent total partition coefficient at blood concentrations of \(1.0\ \mathrm{mg/L}\) and \(10.0\ \mathrm{mg/L}\).
At \(C_B=1.0\ \mathrm{mg/L}\), \(C_{T,u}=0.50(1.0)=0.50\ \mathrm{mg/L}\). Bound concentration is \(12(0.50)/(2.0+0.50)=2.4\ \mathrm{mg/L}\). Thus \(C_{T,\mathrm{tot}}=0.50+2.4=2.9\ \mathrm{mg/L}\), and \(K_{T:b,\mathrm{app}}=2.9/1.0=2.9\).
At \(C_B=10.0\ \mathrm{mg/L}\), \(C_{T,u}=0.50(10.0)=5.0\ \mathrm{mg/L}\). Bound concentration is \(12(5.0)/(2.0+5.0)=8.571\ \mathrm{mg/L}\). Therefore \(C_{T,\mathrm{tot}}=5.0+8.571=13.571\ \mathrm{mg/L}\), and \(K_{T:b,\mathrm{app}}=13.571/10.0=1.357\).
The higher concentration produces a lower apparent coefficient because the binding sites are more occupied. Treating \(K_{T:b,\mathrm{app}}=2.9\) as constant at both concentrations would overpredict total tissue concentration at \(10.0\ \mathrm{mg/L}\).
Choosing and Checking the Binding Representation
The linear capacity-factor approach is most useful when reversible binding is rapid and approximately proportional to free tissue concentration across the modeled range. It preserves a simple effective partition coefficient and a single time constant under constant input. A saturable binding equation is preferable when binding capacity is approached; its apparent partitioning changes with concentration and its local storage capacity changes during a profile. A model with separate free and bound tissue states may be needed if binding or unbinding is not rapid relative to perfusion. That alternative explicitly represents finite-rate exchange, rather than assuming that binding remains at equilibrium at every time.
These representations answer different questions. A larger effective partition coefficient summarizes greater equilibrium total tissue storage under specified conditions. It does not, by itself, establish the mechanism of storage or the rate at which the tissue reaches equilibrium. Likewise, a slow tissue response can result from a large storage capacity in this model, but the binding-adjusted equation should not be used to attribute every slow response to binding: its perfusion-limited and rapid-binding assumptions must be plausible for the tissue and time scale of interest.
A practical check is to keep the concentration basis explicit at each step: use free tissue concentration in the equilibrium relation, use total tissue concentration in the stored amount, and include the binding capacity factor in the amount balance. Check units in the time constant as well: \(V_T/Q\) has units of time, while \(K_{T:b,u}\) and \(R_T\) are dimensionless. If binding is concentration-dependent, report the concentration range and avoid presenting one apparent partition coefficient as universal.
Check Your Understanding
Use the assumptions stated in each question. Solutions are provided after the questions.
- For linear binding with \(K_{\mathrm{bind}}=2.0\) and \(K_{T:b,u}=0.60\), calculate \(R_T\) and \(K_{T:b,\mathrm{eff}}\).
- With fixed blood concentration, what happens to equilibrium free tissue concentration and total tissue concentration when \(R_T\) doubles?
- A tissue has \(V_T=8.0\ \mathrm{L}\), \(Q=2.0\ \mathrm{L/h}\), \(K_{T:b,u}=0.50\), and \(R_T=3.0\). Calculate \(\tau\) and \(t_{1/2}\).
- Why can an apparent partition coefficient measured at low concentration be inappropriate at a much higher concentration when binding is saturable?
Solutions. 1. \(R_T=1+2.0=3.0\), so \(K_{T:b,\mathrm{eff}}=3.0(0.60)=1.8\). 2. With fixed blood concentration, \(C_{T,u}=K_{T:b,u}C_B\) is unchanged; \(C_{T,\mathrm{tot}}=R_TC_{T,u}\) doubles. 3. \(\tau=(8.0\ \mathrm{L})(0.50)(3.0)/(2.0\ \mathrm{L/h})=6.0\ \mathrm{h}\); \(t_{1/2}=\ln(2)(6.0\ \mathrm{h})=4.159\ \mathrm{h}\). 4. At higher concentration, more binding sites are occupied, so bound concentration no longer rises in direct proportion to free concentration. The apparent total partition coefficient therefore changes and may be lower.