Partitioning Intracellular Loss Between Bile and Blood

Why Partition Intracellular Loss?

After drug enters a hepatocyte, it may leave across the canalicular membrane into bile or across the basolateral membrane back toward blood. These routes compete for drug in the same intracellular pool. Their relative clearances determine the partition of intracellular efflux: the share of efflux directed to each destination. This is useful for describing the fate of drug leaving the cell, but it is not by itself a prediction of the fraction of an administered dose excreted in bile.

The calculation is simple only when the two efflux clearances are expressed on a consistent basis and act on the same intracellular concentration. Under those conditions, each route’s flux is its clearance multiplied by that concentration. The common concentration then cancels from the ratio of the two fluxes.

Definition: Let $CL_{\mathrm{can}}$ be the canalicular efflux clearance to bile and $CL_{\mathrm{bas}}$ the basolateral efflux clearance back to blood, both in units of volume per time. For nonnegative clearances with $CL_{\mathrm{can}}+CL_{\mathrm{bas}}>0$, the bile and blood fractions of intracellular efflux are $$ f_{\mathrm{bile}}=\frac{CL_{\mathrm{can}}}{CL_{\mathrm{can}}+CL_{\mathrm{bas}}}, \qquad f_{\mathrm{blood}}=\frac{CL_{\mathrm{bas}}}{CL_{\mathrm{can}}+CL_{\mathrm{bas}}}. $$

These fractions describe competing efflux routes. They are dimensionless, lie between zero and one, and sum to one when the clearances are nonnegative and their sum is positive. If both clearances are zero, the fractions are undefined: there is no modeled efflux to partition.

From Efflux Clearances to Route Fractions

Suppose the modeled intracellular concentration is $C_i$, expressed in amount per volume. For linear efflux, the instantaneous rates leaving through the two routes are

J_{\mathrm{can}}=CL_{\mathrm{can}}C_i, \qquad J_{\mathrm{bas}}=CL_{\mathrm{bas}}C_i.

Each flux has units of amount per time. When there is intracellular drug ($C_i>0$), their ratio is

\frac{J_{\mathrm{can}}}{J_{\mathrm{can}}+J_{\mathrm{bas}}} = \frac{CL_{\mathrm{can}}C_i} {(CL_{\mathrm{can}}+CL_{\mathrm{bas}})C_i} = \frac{CL_{\mathrm{can}}}{CL_{\mathrm{can}}+CL_{\mathrm{bas}}}.

The same cancellation gives the blood fraction. The result relies on both routes acting on the same concentration basis. If one clearance is defined relative to a different concentration, the two values cannot be inserted into this ratio without first reconciling their definitions.

Worked Example: A Direct Bile–Blood Partition

Assume a hypothetical hepatocyte model has $CL_{\mathrm{can}}=18\ \mathrm{mL/min}$ and $CL_{\mathrm{bas}}=6\ \mathrm{mL/min}$. Both clearances are nonnegative, linear, and defined relative to the same intracellular concentration. Find the route fractions.

The total efflux clearance is $18+6=24\ \mathrm{mL/min}$, which is positive, so the fractions are defined:

f_{\mathrm{bile}}=\frac{18}{24}=0.750, \qquad f_{\mathrm{blood}}=\frac{6}{24}=0.250.

Thus, under these assumptions, 75.0% of the modeled intracellular efflux is directed to bile and 25.0% returns to blood. The check is $0.750+0.250=1.000$. These percentages concern drug leaving the intracellular pool, not the administered dose.

For a flux check, if $C_i=0.20\ \mathrm{mg/mL}$, then $J_{\mathrm{can}}=18\ \mathrm{mL/min}\times0.20\ \mathrm{mg/mL}=3.6\ \mathrm{mg/min}$, and $J_{\mathrm{bas}}=6\ \mathrm{mL/min}\times0.20\ \mathrm{mg/mL}=1.2\ \mathrm{mg/min}$. The total is $4.8\ \mathrm{mg/min}$; the route shares are $3.6/4.8=0.750$ and $1.2/4.8=0.250$.

When the Fractions Also Describe Cumulative Export

The clearance ratio gives the instantaneous share of efflux. It also gives the cumulative share of an initial intracellular amount when efflux is linear and constant, the intracellular volume is constant, and no new drug enters the pool after the starting time. The following derivation makes those conditions explicit.

Assumptions for the cumulative result: The intracellular pool is well mixed and has a constant positive volume $V_i$; both clearances are constant, nonnegative, and act on the same concentration; there are no other intracellular loss routes; and $CL_{\mathrm{can}}+CL_{\mathrm{bas}}>0$. Consider an initial amount $A_i(0)>0$ with no further input.

With $C_i=A_i/V_i$, conservation of mass gives the intracellular amount balance

\frac{dA_i}{dt} =-J_{\mathrm{can}}-J_{\mathrm{bas}} =-\frac{CL_{\mathrm{can}}+CL_{\mathrm{bas}}}{V_i}A_i.

Writing $CL_{\mathrm{sum}}=CL_{\mathrm{can}}+CL_{\mathrm{bas}}$, the solution is $A_i(t)=A_i(0)\exp(-CL_{\mathrm{sum}}t/V_i)$. Since $CL_{\mathrm{sum}}>0$, this amount tends to zero as time tends to infinity: all of the initial amount eventually leaves by one of the two modeled routes. Integrating the canalicular flux gives

Q_{\mathrm{can}}(\infty) =\int_0^\infty CL_{\mathrm{can}}\frac{A_i(t)}{V_i}\,dt =A_i(0)\frac{CL_{\mathrm{can}}}{CL_{\mathrm{sum}}}.

The same integration for the basolateral route gives $Q_{\mathrm{bas}}(\infty)=A_i(0)CL_{\mathrm{bas}}/CL_{\mathrm{sum}}$. Dividing each cumulative amount by the positive initial amount shows that the cumulative fractions equal the clearance fractions. If there is continued input, another loss route, changing clearances, or a changing volume, this cumulative conclusion does not automatically follow, even though the instantaneous route ratio may still apply under linear, shared-concentration conditions.

Worked Example: Cumulative Amounts After an Intracellular Dose

Suppose a hypothetical model begins with $A_i(0)=12\ \mathrm{mg}$, $V_i=40\ \mathrm{mL}$, $CL_{\mathrm{can}}=18\ \mathrm{mL/min}$, and $CL_{\mathrm{bas}}=6\ \mathrm{mL/min}$. There is no further input, no other loss, and the clearances and volume remain constant. The clearance sum is $24\ \mathrm{mL/min}>0$, so the cumulative result applies.

The cumulative amount exported to bile is

Q_{\mathrm{can}}(\infty) =12\ \mathrm{mg}\times\frac{18}{18+6} =12\times0.750 =9.00\ \mathrm{mg}.

The amount returned to blood is

Q_{\mathrm{bas}}(\infty) =12\ \mathrm{mg}\times\frac{6}{18+6} =12\times0.250 =3.00\ \mathrm{mg}.

The mass-balance check is $9.00+3.00=12.0\ \mathrm{mg}$, equal to the initial amount. The constant-volume model also has an efflux rate constant $CL_{\mathrm{sum}}/V_i=24/40=0.600\ \mathrm{min}^{-1}$, consistent with eventual depletion of the intracellular pool. This rate describes depletion, while the clearance ratio describes its route partition.

Changes in One Route and a Steady-State Input

The fractions depend on the relative clearances, not on their common scale. Multiplying both clearances by the same positive factor leaves the partition unchanged, though it increases the total rate of intracellular loss at a given concentration. Changing just one clearance can change the partition, but strict directional claims require care at boundary values.

Worked Example: Changing Canalicular Clearance

Consider hypothetical clearances that change from $CL_{\mathrm{can}}=10\ \mathrm{mL/min}$ and $CL_{\mathrm{bas}}=5\ \mathrm{mL/min}$ to $CL_{\mathrm{can}}=5\ \mathrm{mL/min}$ while $CL_{\mathrm{bas}}$ remains $5\ \mathrm{mL/min}$. The fixed basolateral clearance is positive.

Initially, $f_{\mathrm{bile}}=10/(10+5)=0.667$ and $f_{\mathrm{blood}}=5/(10+5)=0.333$. After the change, $f_{\mathrm{bile}}=5/(5+5)=0.500$ and $f_{\mathrm{blood}}=5/(5+5)=0.500$. In this case, decreasing a positive canalicular clearance while the competing clearance remains positive decreases the bile fraction and increases the blood fraction.

The positivity condition matters. If $CL_{\mathrm{bas}}=0$, the bile fraction remains 1 whenever $CL_{\mathrm{can}}>0$, even if $CL_{\mathrm{can}}$ decreases. If $CL_{\mathrm{can}}=0$ as well, the fractions are undefined. Therefore, the general statement is that changing a clearance changes the partition according to the ratio; strict directional changes require the relevant competing clearance to be positive.

At steady state with a constant intracellular input rate $R_{\mathrm{in}}$ and no other loss route, mass balance gives $R_{\mathrm{in}}=CL_{\mathrm{sum}}C_i$. This relationship can be used to calculate the intracellular concentration and the two output rates. It does not change their fractional partition.

Worked Example: Partitioning a Steady-State Efflux Rate

In a hypothetical linear model, let $R_{\mathrm{in}}=4.0\ \mathrm{mg/min}$, $CL_{\mathrm{can}}=12\ \mathrm{mL/min}$, and $CL_{\mathrm{bas}}=8\ \mathrm{mL/min}$. Assume the input is balanced only by these two efflux routes and the system is at steady state. The total clearance is $20\ \mathrm{mL/min}$, so

C_i=\frac{R_{\mathrm{in}}}{CL_{\mathrm{can}}+CL_{\mathrm{bas}}} =\frac{4.0\ \mathrm{mg/min}}{20\ \mathrm{mL/min}} =0.200\ \mathrm{mg/mL}.

The bile flux is $12\ \mathrm{mL/min}\times0.200\ \mathrm{mg/mL}=2.4\ \mathrm{mg/min}$, and the blood flux is $8\ \mathrm{mL/min}\times0.200\ \mathrm{mg/mL}=1.6\ \mathrm{mg/min}$. Their sum, $4.0\ \mathrm{mg/min}$, matches the input. The fractions are $2.4/4.0=0.600$ to bile and $1.6/4.0=0.400$ to blood. This partition applies to the modeled efflux at steady state; it does not, without additional model information, identify the fraction of an administered dose reaching either destination.

Diagnostics, Alternatives, and Limitations

A reliable calculation begins by checking that the two clearances refer to the same intracellular pool and the same concentration basis. Their units must match, and the total must be positive before fractions are calculated. A useful numerical check is that both fractions are between zero and one and add to one. When fluxes are available, independently calculate $J_{\mathrm{can}}$ and $J_{\mathrm{bas}}$ and verify that their ratios agree with the clearance fractions.

The clearance-ratio calculation is a compact alternative to repeatedly solving the intracellular mass-balance equation when the question is only the instantaneous route share and the assumptions hold. A full mass-balance model is needed to predict intracellular amounts or concentrations over time, or to handle changing input, changing volume, or additional routes. If efflux is nonlinear or saturable, each route’s rate may depend on concentration in a different way; a single constant clearance ratio may then fail to describe the partition over a range of concentrations. At a specified concentration, route fractions can instead be calculated from the actual nonnegative fluxes, provided their sum is positive.

The approach also has a scope limitation: partitioning two intracellular loss clearances is not the same as calculating whole-liver clearance. Uptake from blood, other elimination processes, and physiological scaling may influence how much drug reaches the intracellular pool or how much leaves the liver overall. The ratio answers the narrower question of where modeled intracellular efflux goes.

Using the wrong denominator. Dividing $CL_{\mathrm{can}}$ by itself gives 1, but ignores competition. Use the sum of both competing clearances.
Ignoring the zero-sum case. If both clearances are zero, do not report fractions: the denominator is zero and no modeled efflux occurs.
Confusing a route fraction with a dose fraction. A 75% bile share of intracellular efflux does not establish that 75% of an administered dose appears in bile.
Assuming a fixed partition for nonlinear fluxes. If route rates do not remain proportional to the same intracellular concentration, calculate the flux shares at the relevant state or use a fuller model.

Key takeaway: For two nonnegative, linear efflux clearances acting on the same intracellular concentration, divide each clearance by their positive sum to obtain its share of intracellular efflux. Interpret that share within the model’s boundary conditions, not as a whole-dose or whole-liver fraction.

Check Your Understanding

Answer each question, then check the solutions below.

If $CL_{\mathrm{can}}=9\ \mathrm{mL/min}$ and $CL_{\mathrm{bas}}=3\ \mathrm{mL/min}$, what are the bile and blood fractions?
What can be concluded about the fractions when both clearances are zero?
Under what conditions do the clearance fractions also equal the cumulative fractions of an initial intracellular amount?
If $CL_{\mathrm{can}}$ falls while $CL_{\mathrm{bas}}=0$, must the bile fraction decrease?
At steady state, an input of $6\ \mathrm{mg/min}$ is balanced by $CL_{\mathrm{can}}=15\ \mathrm{mL/min}$ and $CL_{\mathrm{bas}}=5\ \mathrm{mL/min}$. Find the intracellular concentration and both efflux rates.

Solutions. (1) The sum is $12\ \mathrm{mL/min}$, so $f_{\mathrm{bile}}=9/12=0.750$ and $f_{\mathrm{blood}}=3/12=0.250$. (2) The fractions are undefined because the denominator is zero; neither route has modeled efflux. (3) The pool must have constant positive volume, constant nonnegative linear clearances acting on the same concentration, no other loss routes, no continued input after an initial positive amount, and a positive clearance sum. Then the pool ultimately empties and integrated route amounts have the same shares as the clearances. (4) No. While $CL_{\mathrm{can}}>0$, the bile fraction remains 1; if it reaches zero too, the fractions become undefined. (5) The total clearance is $20\ \mathrm{mL/min}$, so $C_i=6/20=0.300\ \mathrm{mg/mL}$. The bile rate is $15\times0.300=4.5\ \mathrm{mg/min}$, and the blood rate is $5\times0.300=1.5\ \mathrm{mg/min}$; the rates sum to the $6\ \mathrm{mg/min}$ input.