Hepatic Availability From First-Pass Extraction

From Extraction to Hepatic Availability

Hepatic extraction describes how much drug the liver removes during a passage; hepatic availability describes the complementary fraction that escapes that passage unchanged. The distinction matters in PBPK predictions of first-pass loss: a model's clearance prediction can be translated into a survival fraction, but the translation must use the same liver model and consistent blood-based inputs.

Here, hepatic availability refers to one passage through the liver. It does not by itself include absorption from the gut, intestinal metabolism, or other losses before or after that passage. When those processes are represented separately, their fractions may be combined as appropriate; hepatic availability should not be counted twice.

Definition: For hepatic inlet concentration \(C_{\mathrm{in}}\) and outlet concentration \(C_{\mathrm{out}}\), the hepatic extraction ratio is \(E_H=(C_{\mathrm{in}}-C_{\mathrm{out}})/C_{\mathrm{in}}\). The hepatic availability is the fraction leaving the liver unchanged, \(F_H=C_{\mathrm{out}}/C_{\mathrm{in}}=1-E_H\), for positive inlet concentration.

This identity follows directly by rearranging the definition: \(E_H=1-C_{\mathrm{out}}/C_{\mathrm{in}}\), so \(1-E_H=C_{\mathrm{out}}/C_{\mathrm{in}}\). Thus, once a model supplies the extraction ratio, availability is not a second independent estimate. It is its complement.

Availability in the Three Liver Models

Use hepatic blood flow \(Q\), unbound fraction \(f_u\), and unbound intrinsic clearance \(CL_{\mathrm{int}}\), expressed on a consistent blood basis. Define the dimensionless intrinsic-clearance-to-flow ratio \(R_N=f_uCL_{\mathrm{int}}/Q\). Assume \(Q>0\), \(CL_{\mathrm{int}}\geq 0\), and \(0<f_u\leq1\). Under these conditions \(R_N\geq0\), and the predicted availability should lie between zero and one.

In the well-stirred model, the liver is treated as a single, completely mixed compartment. Its steady-state clearance is \(CL_H=Q R_N/(1+R_N)\). Dividing by flow gives \(E_H=CL_H/Q=R_N/(1+R_N)\). Therefore:

F_H^{\mathrm{WS}}=1-E_H=\frac{1}{1+R_N}=\frac{Q}{Q+f_uCL_{\mathrm{int}}}.

In the parallel-tube model, blood moves along tubes with no axial mixing between them, and drug is removed continuously along the path. The concentration decreases exponentially with distance. At the outlet, the surviving fraction is the exponential of the negative total removal-to-flow ratio:

F_H^{\mathrm{PT}}=\exp(-R_N) =\exp\left(-\frac{f_uCL_{\mathrm{int}}}{Q}\right), \qquad E_H^{\mathrm{PT}}=1-\exp(-R_N).

The dispersion model allows axial mixing as well as forward flow and removal. It uses a dimensionless dispersion number \(D_N>0\), which represents the degree of axial dispersion relative to convection. In the usual steady-state, one-dimensional formulation, let \(x\) be distance along a normalized liver length, and let \(C(x)\) be concentration. The model equation and boundary conditions are:

D_N\frac{d^2C}{dx^2}-\frac{dC}{dx}-R_NC=0,\qquad C(0)-D_N C'(0)=C_{\mathrm{in}},\qquad C'(1)=0.

The inlet condition specifies the incoming convective-dispersive flux, and the outlet condition specifies zero concentration gradient at the exit. This is a steady-state model with uniform first-order removal, constant flow and dispersion, and no additional sources or sinks within the liver. Setting \(a=\sqrt{1+4D_NR_N}\), the characteristic roots of the differential equation are \((1+a)/(2D_N)\) and \((1-a)/(2D_N)\). Substituting the corresponding exponential solution into the boundary conditions gives the outlet-to-inlet ratio:

F_H^{\mathrm{D}}= \frac{4a\exp\left(\frac{1}{2D_N}\right)} {(1+a)^2\exp\left(\frac{a}{2D_N}\right) -(1-a)^2\exp\left(-\frac{a}{2D_N}\right)}, \qquad E_H^{\mathrm{D}}=1-F_H^{\mathrm{D}}.

This is the dispersion-model availability, not a clearance formula to be inserted into another model. The derivation also gives useful checks: if \(R_N=0\), then \(a=1\) and the expression reduces to \(F_H^{\mathrm{D}}=1\), as required when there is no removal. If dispersion tends to zero while other inputs are held fixed, the expression approaches \(\exp(-R_N)\), the parallel-tube result.

Worked Model Comparison

Worked Example: Same Inputs, Three Availability Predictions

Consider hypothetical liver inputs \(Q=90.0\ \mathrm{L/h}\) and \(f_uCL_{\mathrm{int}}=180\ \mathrm{L/h}\). For the dispersion calculation, suppose \(D_N=0.200\). These are illustrative values, not measurements from a study. First calculate the shared dimensionless ratio:

R_N=\frac{180\ \mathrm{L/h}}{90.0\ \mathrm{L/h}}=2.00.

For the well-stirred model:

F_H^{\mathrm{WS}}=\frac{1}{1+2.00}=0.3333.

For the parallel-tube model:

F_H^{\mathrm{PT}}=\exp(-2.00)=0.1353.

For the dispersion model, \(a=\sqrt{1+4(0.200)(2.00)}=\sqrt{2.60}=1.612\). Substitution into the dispersion expression gives:

F_H^{\mathrm{D}}= \frac{4(1.612)\exp(2.50)} {(2.612)^2\exp(4.031)-(0.612)^2\exp(-4.031)} \approx 0.2044.

Under these inputs, the well-stirred model predicts that about 33.33% of the entering drug escapes hepatic passage, the parallel-tube model about 13.53%, and the dispersion model about 20.44%. These are differences in model predictions, not evidence that one calculation is arithmetically inconsistent with another: the models represent different mixing patterns.

Connecting Hepatic Availability to First-Pass Survival

Hepatic availability is often one component of an overall fraction reaching systemic circulation. If a route model separately specifies a fraction absorbed \(F_a\), an intestinal fraction escaping metabolism \(F_g\), and hepatic availability \(F_H\), and if these sequential fractions are defined on compatible bases, their product is the fraction reaching systemic circulation:

$$ F=F_aF_gF_H. $$

This multiplication is appropriate only when each factor represents a distinct sequential loss. It is not appropriate to multiply by a hepatic fraction already embedded in another model output, or to apply an oral first-pass factor to a route that bypasses the liver on initial entry.

Worked Example: Model Choice in a Sequential Fraction

Continue with the hypothetical inputs above. Suppose \(F_a=0.80\) and \(F_g=0.90\), with both processes represented separately from hepatic passage. Their product before hepatic passage is \(0.80\times0.90=0.72\). Using the well-stirred prediction gives:

F_{\mathrm{WS}}=(0.80)(0.90)\left(\frac{1}{3}\right)=0.2400.

For the parallel-tube prediction, retain the unrounded exponential until the final multiplication:

F_{\mathrm{PT}}=(0.80)(0.90)\exp(-2.00) =0.72\exp(-2.00)\approx 0.09744.

For the dispersion prediction, using the displayed rounded availability:

F_{\mathrm{D}}=(0.80)(0.90)(0.2044) =(0.72)(0.2044)=0.147168\approx0.1472.

The overall predicted fractions are therefore 0.2400, 0.09744, and 0.1472, respectively. They differ because the predicted hepatic survival fractions differ; the values of \(F_a\) and \(F_g\) were held fixed. If the displayed rounded parallel-tube availability \(0.1353\) were used instead, the product would be \(0.72(0.1353)=0.09742\). Keeping the exponential unrounded until the final result yields the reported \(0.09744\).

Worked Example: Converting Inlet and Outlet Concentrations

Suppose a hypothetical single-pass measurement has \(C_{\mathrm{in}}=4.00\ \mathrm{mg/L}\) and \(C_{\mathrm{out}}=1.40\ \mathrm{mg/L}\). These concentrations are positive and refer to inlet and outlet on the same basis, so the extraction ratio is:

E_H=\frac{4.00-1.40\ \mathrm{mg/L}}{4.00\ \mathrm{mg/L}} =\frac{2.60}{4.00}=0.650.

The units cancel. The hepatic availability is the complementary outlet fraction:

F_H=\frac{1.40\ \mathrm{mg/L}}{4.00\ \mathrm{mg/L}}=0.350=1-0.650.

Thus, 35.0% of the drug entering the liver exits unchanged in this example. This concentration-ratio calculation defines the observed single-pass fraction; it does not identify which liver model generated it.

Checks, Alternatives, and Limitations

Before interpreting a result, verify that flow and clearance quantities use compatible units and blood conventions, and that \(R_N\) is dimensionless. Check that \(0\leq F_H\leq1\) and \(0\leq E_H\leq1\) for nonnegative removal. At \(R_N=0\), every model should give \(F_H=1\). Increasing removal at fixed flow should not increase availability. For the dispersion model, also check that \(D_N\) is positive and that the result approaches the parallel-tube prediction as dispersion becomes negligible.

An estimate outside the range zero to one can indicate inconsistent units, an incorrect sign in an exponent, a transcription error, or misuse of a formula. For example, calculating \(E_H\) and then reporting it as \(F_H\) reverses the meaning: with \(E_H=0.65\), the escaping fraction is \(0.35\), not \(0.65\). For products of fractions, state whether inputs are rounded; premature rounding can produce small differences in the last reported digits.

The models are alternatives for describing intrahepatic mixing and concentration profiles. The well-stirred model is compact and assumes complete mixing; the parallel-tube model assumes no axial mixing; the dispersion model adds axial mixing through \(D_N\). Their availability predictions need not match even when \(Q\) and \(f_uCL_{\mathrm{int}}\) agree, because their structural assumptions differ. A measured concentration ratio can provide an availability directly, while a model-based prediction requires selecting and parameterizing a model. Neither a plausible range check nor agreement with one observation proves that the chosen structure is uniquely correct.

All three expressions describe a specified hepatic passage under their model assumptions. They do not represent time-varying flow or intrinsic clearance, saturation, active transport, recirculation, or additional hepatic processes unless those are explicitly included elsewhere. The dispersion formula also depends on a defined dispersion number and its boundary-condition convention; values from incompatible conventions should not be substituted without conversion.

Key takeaway: Hepatic availability is \(1-E_H\), but the value of \(E_H\) depends on the chosen liver model. Use one model consistently, preserve precision through intermediate calculations, and keep hepatic survival distinct from other route-specific fractions.

Check Your Understanding

Answer each question, then compare with the solutions below.

If \(R_N=1.00\), what are the well-stirred and parallel-tube hepatic availability predictions?
A hypothetical liver has \(C_{\mathrm{in}}=5.00\ \mathrm{mg/L}\) and \(C_{\mathrm{out}}=2.00\ \mathrm{mg/L}\). Calculate \(E_H\) and \(F_H\).
Why can the three model predictions differ when \(Q\) and \(f_uCL_{\mathrm{int}}\) are identical?
Suppose \(F_a=0.75\), \(F_g=0.80\), and a selected model gives \(F_H=0.40\). Calculate the overall fraction under the sequential-fraction assumptions.

Solutions. (1) \(F_H^{\mathrm{WS}}=1/(1+1)=0.5000\); \(F_H^{\mathrm{PT}}=\exp(-1)=0.3679\). (2) \(E_H=(5.00-2.00)/5.00=0.600\), and \(F_H=2.00/5.00=0.400=1-E_H\). (3) The models impose different assumptions about mixing and concentration profiles within the liver, so identical flow and intrinsic-clearance inputs do not require identical outlet fractions. (4) \(F=(0.75)(0.80)(0.40)=0.240\), or 24.0% reaching systemic circulation under the stated sequential-fraction assumptions.