Incorporating Hemodialysis into Impaired-Renal PBPK

Why dialysis must be represented as a timed process

In renal impairment, a drug may still be eliminated by residual renal function and by nonrenal pathways. Hemodialysis adds another removal pathway, but only while treatment is operating. A model that replaces renal clearance with a single “dialysis clearance” can therefore miss both continuing elimination between sessions and the sharp change in elimination during treatment. The key modeling task is to represent dialysis as a scheduled, concentration-dependent sink that operates alongside the patient’s other clearance pathways.

The formulation below uses dialyzer inlet and outlet concentrations and the treatment blood or plasma flow to estimate removal. It then inserts that removal into a PBPK model at the appropriate time. The calculations are simplest when concentrations at the dialyzer inlet correspond to the concentration in a well-mixed, accessible central compartment. If tissue distribution or blood-flow limitations make that approximation poor, dialysis removal should instead be linked to a modeled vascular compartment and the model should represent redistribution.

For example, if whole-blood concentrations are used, \(Q\) must be whole-blood flow through the dialyzer. If plasma concentrations are used, use plasma flow, or convert consistently. Mixing plasma concentration with whole-blood flow without a justified conversion can produce a clearance with the wrong scale. In practice, inlet and outlet samples should be paired closely in time and collected under stable treatment conditions; otherwise the ratio may reflect changing concentrations or sampling variation as well as extraction.

Adding the dialysis sink to a PBPK model

Let \(I_{\mathrm{HD}}(t)\) equal 1 during a dialysis session and 0 otherwise. In a PBPK model, dialysis is represented as removal from the modeled compartment supplying the dialyzer. If that compartment has concentration \(C_{\mathrm{access}}(t)\), the removal rate is \(Q(t)E(t)C_{\mathrm{access}}(t)\). The schedule, flow, and extraction can be time-varying. For a multi-compartment model, this sink belongs in the amount balance for the relevant vascular or central compartment, while intercompartmental exchange and other elimination processes remain in their own equations.

A useful simplified case assumes a single well-mixed accessible compartment with amount \(A(t)\), volume \(V\), and concentration \(C(t)=A(t)/V\). Suppose residual renal clearance \(CL_{\mathrm{renal,res}}\) and nonrenal clearance \(CL_{\mathrm{nonrenal}}\) are linear with respect to this same concentration. With no input during the interval, the amount balance is:

This formulation assumes that the clearances are expressed on a common concentration basis, the parameters are constant within each modeled time segment, and dialysis does not itself change distribution volume or binding during that segment. If renal or nonrenal elimination is nonlinear, or depends on unbound rather than total concentration, use the appropriate rate law instead of treating it as a constant clearance multiplied by total concentration. Likewise, do not include a renal clearance that has already been removed or reduced in the impaired-renal parameterization.

Deriving the concentration change over a session

For the one-compartment case, suppose the input is zero and all clearances, flow, and extraction are constant during a treatment interval of length \(T\). Since \(A=VC\), the amount equation becomes a first-order differential equation in concentration. Dividing by \(V\) gives the concentration decline rate:

Separating variables and integrating from the start concentration \(C_0\) to the concentration \(C(T)\) yields \(\ln[C(T)/C_0]=-(CL_{\mathrm{renal,res}}+CL_{\mathrm{nonrenal}}+QE)T/V\). Exponentiating gives:

The same expression applies between sessions with \(QE=0\). For multiple sessions or changing inputs, apply the balance piecewise at the actual session start and stop times, or integrate the full PBPK differential equations numerically. A concentration measured after treatment may not be the end-of-session concentration: tissue-to-plasma redistribution can cause rebound, which a single well-mixed compartment cannot represent.

Worked concentration predictions with treatment timing

Assumptions, diagnostics, and alternatives

A useful dialysis representation requires a credible session schedule, compatible flow and concentration measurements, and an extraction estimate that is physically and analytically plausible. For nonnegative concentrations, require \(C_{\mathrm{in}}>0\) and \(C_{\mathrm{out}}\leq C_{\mathrm{in}}\) before interpreting the usual extraction ratio as lying between 0 and 1. If \(C_{\mathrm{in}}=C_{\mathrm{out}}=0\), the ratio is undefined, not zero. A negative estimate or one above 1 calls for review of assay error, sampling, units, sample timing, and the extraction definition; it should not be silently clipped to the interval from 0 to 1.

Compare predicted and observed concentrations at times that identify the phases of interest: before dialysis, near session end, and after a defined post-treatment interval. A systematic mismatch that grows during treatment may indicate a poor extraction estimate, changing flow, or an incorrectly specified dialyzer-accessible concentration. A good end-session fit followed by a poor post-session fit may point to distributional rebound or an incorrect off-dialysis elimination rate. These patterns are diagnostic clues, not proof of a single mechanism.

A time-averaged dialysis clearance can sometimes approximate average exposure over a repeated schedule, but it loses the concentration changes associated with treatment starts and stops. It is most defensible when the question concerns a sufficiently long-term average and the schedule is regular; it is unsuitable for predicting immediate postdialysis concentrations or the effect of a dose close to treatment. A time-varying clearance in a one-compartment model preserves the schedule but still assumes instant mixing. A multi-compartment PBPK approach is more appropriate when tissue distribution, blood-to-plasma differences, or rebound materially affect concentrations. Directly representing dialyzer inlet and outlet compartments can be useful when flow and extraction vary substantially or when dialyzer behavior is the modeling focus.

Dialysis removal should be interpreted as one pathway in the amount balance, not as a replacement for residual renal or nonrenal elimination. The principal limitations of the simple calculation are its dependence on representative extraction measurements and its assumptions about mixing, linearity, and concentration basis. The schedule-aware formulation makes those assumptions explicit and provides a starting point for adding dialysis to a more detailed PBPK model.