Flow Sensitivity in the Three Liver Models

Why Flow Sensitivity Is Not the Same in Every Liver Model

A liver model predicts more than a clearance value at one blood flow. It also implies how clearance changes when hepatic blood flow changes while the drug’s effective intrinsic capacity is held fixed. That response matters when interpreting flow perturbations, but it is model-dependent: the well-stirred model treats the liver as a single mixed compartment, the parallel-tube model represents concentration change along flow paths, and the dispersion model additionally allows axial mixing.

The relevant comparison is a flow sensitivity, not simply a comparison of clearances at different flows. A useful dimensionless measure is the logarithmic elasticity of hepatic clearance with respect to hepatic blood flow. It describes the local percentage change in clearance associated with a percentage change in flow. For the dispersion model, the calculation must also specify whether its dimensionless dispersion number is held fixed or changes as flow changes.

Definition: Let $Q$ be hepatic blood flow and $CL_H$ hepatic clearance. The local flow elasticity is $$ S_Q=\frac{d\ln(CL_H)}{d\ln(Q)} =\frac{Q}{CL_H}\frac{dCL_H}{dQ}. $$ For example, $S_Q=0.5$ means that, near the specified operating point and under the stated assumptions, a 1% increase in $Q$ is associated with approximately a 0.5% increase in $CL_H$.

These sensitivities are model predictions under a controlled mathematical perturbation. The main assumption is that the effective intrinsic hepatic clearance capacity, denoted $K$, does not change as $Q$ changes. Other determinants of hepatic clearance are also held fixed unless stated otherwise. In particular, a flow change in a real physiological system might coincide with changes in enzyme activity, binding, transporter activity, or organ state; those effects are not represented by the flow elasticity calculated here.

Well-Stirred and Parallel-Tube Sensitivities

Use $E$ for the hepatic extraction ratio, so $CL_H=QE$. In these derivations, define the dimensionless capacity-to-flow ratio $r=K/Q$. The earlier material on extraction ratios establishes the interpretation of $E$; here the focus is how $E$, and therefore clearance, responds to $Q$ when $K$ is fixed.

Model results: Under the well-stirred model, $E=r/(1+r)$, giving $CL_H=QK/(Q+K)$. Under the parallel-tube model, $E=1-e^{-r}$, giving $CL_H=Q(1-e^{-r})$. In both cases $r=K/Q$.

For the well-stirred model, substituting $CL_H=QK/(Q+K)$ into the definition of $S_Q$, with $K$ fixed, gives

\frac{dCL_H}{dQ} =\frac{K(Q+K)-QK}{(Q+K)^2} =\frac{K^2}{(Q+K)^2}, \qquad S_Q=\frac{Q}{QK/(Q+K)}\frac{K^2}{(Q+K)^2} =\frac{K}{Q+K} =1-E.

Thus the well-stirred model’s local flow elasticity is between zero and one for positive $Q$ and $K$. As extraction becomes high, $E$ approaches one and the predicted elasticity approaches zero: clearance approaches the flow ceiling. At low extraction, $E$ is small and the elasticity approaches one.

For the parallel-tube model, differentiate with respect to $Q$ through $r=K/Q$. Since $dr/dQ=-r/Q$, the extraction ratio has derivative $dE/dQ=-r e^{-r}/Q$. Therefore,

S_Q =1+\frac{Q}{E}\frac{dE}{dQ} =1-\frac{r e^{-r}}{1-e^{-r}} =1-\frac{r}{e^r-1}.

When $r$ is small, the exponential expansion $e^r-1\approx r$ makes $S_Q$ approach zero: clearance is approximately $K$, so a small flow change has little effect. When $r$ is large, $r/(e^r-1)$ approaches zero, and $S_Q$ approaches one: clearance is approaching $Q$, so it is increasingly flow-sensitive.

Worked Example: Comparing Well-Stirred and Parallel-Tube Elasticities

Suppose, hypothetically, that $Q=100$ L/h and $K=100$ L/h. Thus $r=K/Q=1$. Calculate the clearance and local flow elasticity under both models.

Well-stirred model. The extraction ratio is $E=r/(1+r)=1/2=0.5000$, so $CL_H=QE=100(0.5000)=50.00$ L/h. Its elasticity is $S_Q=1-E=0.5000$.

Parallel-tube model. The extraction ratio is $E=1-e^{-1}=1-0.3679=0.6321$. Therefore $CL_H=100(0.6321)=63.21$ L/h. Its elasticity is $S_Q=1-1/(e^1-1)=1-1/1.7183=0.4180$.

At this operating point, the parallel-tube model predicts higher clearance but lower local flow elasticity than the well-stirred model. These are separate comparisons: a larger clearance does not necessarily imply a larger percentage response to flow.

Dispersion-Model Sensitivity Requires a Dispersion Assumption

In the dispersion model, let $D$ denote the dimensionless dispersion number and $R(r,D)$ the ratio of outlet to inlet concentration. The extraction ratio is $E=1-R$, so $CL_H=Q[1-R(r,D)]$. For the standard axial-dispersion expression, define $a=\sqrt{1+4Dr}$. One form of the outlet ratio is

R(r,D)= \frac{4a\exp\!\left(\frac{1-a}{2D}\right)} {(1+a)^2-(1-a)^2\exp\!\left(-\frac{a}{D}\right)}.

This expression applies to the specified axial-dispersion model and its associated boundary conditions, with positive $r$ and $D$. Its limiting behavior connects to other liver-model forms, but the sensitivity calculation must respect how $D$ is treated during the flow perturbation. Holding $D$ fixed is a useful controlled mathematical comparison. It is not automatically the physical consequence of changing flow in a fixed organ.

Differentiate $CL_H=Q(1-R)$ by the product and chain rules. Because $r=K/Q$, $dr/dQ=-r/Q$. Allowing $D$ to depend on $Q$ gives the general elasticity:

\frac{dCL_H}{dQ} =(1-R)-Q\left( \frac{\partial R}{\partial r}\frac{dr}{dQ} +\frac{\partial R}{\partial D}\frac{dD}{dQ} \right) =(1-R)+r\frac{\partial R}{\partial r} -Q\frac{\partial R}{\partial D}\frac{dD}{dQ},

S_Q =1+\frac{r}{1-R}\frac{\partial R}{\partial r} -\frac{Q}{1-R}\frac{\partial R}{\partial D}\frac{dD}{dQ}.

The minus sign on the final term follows directly from differentiating the extraction factor $1-R$. If $D$ is held fixed, $dD/dQ=0$, and that term vanishes. If the axial dispersion coefficient and liver geometry are held fixed while flow changes, the usual dimensionless definition makes $D$ inversely proportional to flow, so $dD/dQ=-D/Q$. In that case the final contribution becomes $+D(\partial R/\partial D)/(1-R)$. These are different perturbation assumptions and need not yield the same sensitivity.

Choose what is held fixed.
Specify whether $D$ remains fixed or follows an assumed flow-scaling relationship. Also hold $K$ fixed for a flow-only sensitivity.

Differentiate the extraction factor.
Use $CL_H=Q(1-R)$, with $r=K/Q$, and include both $r$ and $D$ derivatives if $D$ changes with $Q$.

Interpret the elasticity locally.
Report the operating point and assumptions; $S_Q$ describes a local percentage response, not a universal constant for the model.

Worked Dispersion Calculations

Worked Example: Dispersion Sensitivity With Fixed Dispersion Number

For a hypothetical case, take $r=1$ and $D=0.2000$, and hold $D$ fixed during the flow perturbation. Calculate $E$ and the local elasticity. The derivatives below can be obtained by differentiating the displayed $R(r,D)$ expression.

First, $a=\sqrt{1+4(0.2000)(1)}=\sqrt{1.8}=1.3416$. Substitution into the outlet-ratio formula gives $R\approx 0.4166$, so $E=1-R\approx0.5834$.

At this point, differentiating the formula with respect to $r$ while holding $D=0.2000$ gives $\partial R/\partial r\approx-0.3240$. Since $dD/dQ=0$, the general result reduces to $S_Q=1+[r/(1-R)](\partial R/\partial r)$. Thus $S_Q=1+[1/0.5834](-0.3240)\approx0.4446$.

Under this specified fixed-$D$ comparison, a 1% increase in flow near this operating point predicts approximately a 0.444% increase in clearance. The derivative is local and depends on the chosen $r$ and $D$.

Worked Example: A Flow Change When Dispersion Number Scales With Flow

Use the hypothetical starting point $Q=100$ L/h, $K=100$ L/h, and $D=0.2000$, so $r=1$. Now double flow to $200$ L/h, keeping $K$, geometry, and the axial dispersion coefficient fixed. Under the stated scaling, $D$ halves to $0.1000$, and $r$ also halves to $0.5000$.

At the starting point, the preceding calculation gave $E\approx0.5834$, hence $CL_H\approx100(0.5834)=58.34$ L/h. At the doubled flow, $a=\sqrt{1+4(0.1000)(0.5000)}=\sqrt{1.2}=1.0954$. Substitution into the dispersion expression gives $R\approx0.6192$, so $E\approx0.3808$ and $CL_H\approx200(0.3808)=76.16$ L/h.

The clearance ratio is $76.16/58.34\approx1.305$: clearance increases by about 30.5% when flow doubles. The corresponding two-point logarithmic elasticity is $\ln(76.16/58.34)/\ln(200/100)\approx0.3845$. This is an average elasticity over the twofold change, not the local derivative at either endpoint. It differs from the fixed-$D$ local elasticity because $D$ changes with flow in this example.

Interpreting, Comparing, and Checking Sensitivities

For the hypothetical starting values $Q=K=100$ L/h, the well-stirred and parallel-tube models give local elasticities of 0.500 and 0.418, respectively. The dispersion model at $D=0.2000$ gives approximately 0.444 when $D$ is fixed. If instead dispersion number scales inversely with flow, the twofold perturbation in the worked example gives an average elasticity of approximately 0.385. These values illustrate why model, operating point, and perturbation assumption must accompany a reported sensitivity.

A useful diagnostic is to calculate clearance at nearby flow values as well as evaluating a derivative. For a sufficiently small change, the percentage change in clearance should be close to $S_Q$ times the percentage change in flow. A disagreement may indicate an arithmetic or differentiation error, an overly large perturbation for a local approximation, or inconsistent treatment of $D$. For dispersion calculations, check that $0\leq E\leq1$ and that clearance does not exceed $Q$ under the standard extraction interpretation.

The models offer alternative representations, not interchangeable sensitivity formulas. The well-stirred and parallel-tube elasticities follow directly from their respective extraction equations. The dispersion model adds a structural parameter whose flow dependence must be specified. A fixed-$D$ sensitivity is appropriate when the goal is to isolate the effect of changing $r$ at a fixed dimensionless dispersion number. An inverse-flow scaling is appropriate only under its stated physical assumptions. Neither assumption should be silently substituted for the other.

A common mistake is to differentiate only $r=K/Q$ in the dispersion model even when $D$ changes with $Q$, or to use a plus sign before the general $D$-derivative term. The correct chain rule is $-Q(\partial R/\partial D)(dD/dQ)$ in $dCL_H/dQ$. When $dD/dQ=-D/Q$, this contribution is positive in the elasticity numerator; when $D$ is fixed, it is zero. The sign and whether the term is present both follow from the explicitly chosen $D(Q)$.

Finally, a flow elasticity is not by itself evidence that flow caused an observed clearance change in a patient. The calculation assumes other determinants are constant and describes the specified model, not a complete account of physiology. Large perturbations, changing intrinsic capacity, or measurement uncertainty may require direct model evaluation rather than extrapolation from a local derivative.

Check Your Understanding

Answer each question using the definitions and assumptions in this tutorial. Brief solutions follow.

For the well-stirred model, what is $S_Q$ when $E=0.80$, with effective intrinsic capacity held fixed?
At $r=1$, calculate the parallel-tube flow elasticity using $S_Q=1-r/(e^r-1)$.
In the dispersion model, why is it necessary to state whether $D$ changes with $Q$?
If $dD/dQ=-D/Q$, what does the general dispersion elasticity’s final term become?
Does a model with a higher clearance at one flow necessarily have a higher flow elasticity there?

Solutions. (1) $S_Q=1-E=1-0.80=0.20$. (2) $S_Q=1-1/(e-1)=1-1/1.7183\approx0.4180$. (3) Because $CL_H=Q[1-R(r,D)]$, so a flow-dependent $D$ contributes to the derivative through $\partial R/\partial D$. (4) It becomes $+D(\partial R/\partial D)/(1-R)$, since $-Q(dD/dQ)=D$. (5) No. Clearance level and percentage response are distinct quantities; the worked well-stirred and parallel-tube calculations illustrate this distinction.