Drug Dosing Around Hemodialysis Sessions

Why Timing Around Hemodialysis Matters

For a drug cleared partly by the kidneys, a hemodialysis session can change exposure in two ways: it can remove drug during treatment, and it can change when a dose is given relative to that removal. The concentration just before a session, the concentration at its end, and concentrations measured during the following hours therefore answer different questions. Interpreting them requires the exact sampling times and dosing history, not just a comparison of two values.

This case study develops a deliberately simple starting model. It separates ordinary elimination from additional elimination during dialysis, then uses that model to compare supplementation schedules. The calculations are useful for organizing evidence and designing simulations; they do not, by themselves, establish a clinical dose recommendation. Patient-specific decisions also depend on the drug, dialysis prescription, unbound fraction, clinical response, and uncertainty in the data and model.

Definition: Let \(C(t)\) be the measured drug concentration at time \(t\). Let \(k_{\mathrm{el}}\) be the first-order elimination rate constant when dialysis is not occurring, in \(\mathrm{h}^{-1}\), and \(k_{\mathrm{HD}}\) the additional first-order rate constant during hemodialysis, also in \(\mathrm{h}^{-1}\). In a one-compartment model with no dose input during a dialysis session of duration \(T\), the concentration changes from \(C_{\mathrm{pre}}\) to \(C_{\mathrm{post}}\) according to \(C_{\mathrm{post}}=C_{\mathrm{pre}}\exp[-(k_{\mathrm{el}}+k_{\mathrm{HD}})T]\). This assumes constant rates, a well-mixed central compartment, and no important redistribution during the interval.

Estimating the Dialysis-Related Component

A decline from the pre-dialysis to the post-dialysis concentration is not all attributable to the dialysis machine. Ordinary elimination would also occur over those hours. If \(k_{\mathrm{el}}\) is known or estimated from suitable non-dialysis data, rearranging the one-compartment equation gives an estimate of the additional dialysis-related rate:

\widehat{k}_{\mathrm{HD}} = \frac{\ln(C_{\mathrm{pre}}/C_{\mathrm{post}})}{T} - k_{\mathrm{el}}.

This estimate is meaningful only when the samples represent the same modelled compartment and interval, there is no unaccounted dose or infusion, and concentrations are quantifiable and recorded with reliable times. It should be nonnegative under the proposed model. A negative estimate may reflect measurement variability, an unsuitable estimate of \(k_{\mathrm{el}}\), timing errors, or model failure; it is not evidence that dialysis adds drug to the body.

Worked Example: Estimating Additional Elimination During Dialysis

Suppose a hypothetical patient has a pre-dialysis concentration of \(8.00\ \mathrm{mg/L}\) and an immediate end-of-session concentration of \(3.50\ \mathrm{mg/L}\) after \(4.00\ \mathrm{h}\) of dialysis. A separate non-dialysis profile supports \(k_{\mathrm{el}}=0.100\ \mathrm{h}^{-1}\). Assume no dose was given during the session and use the one-compartment model.

The total rate implied by the session concentrations is:

\frac{\ln(8.00/3.50)}{4.00} = \frac{\ln(2.2857)}{4.00} = \frac{0.82668}{4.00} = 0.20667\ \mathrm{h}^{-1}.

Subtracting ordinary elimination gives:

\widehat{k}_{\mathrm{HD}} = 0.20667-0.100 = 0.10667\ \mathrm{h}^{-1}.

The observed concentration fell by \((8.00-3.50)/8.00=0.5625\), or \(56.25\%\). That is the total decline over the session, not the dialysis-specific fraction: ordinary elimination contributes to it. Under the model, the proportional reduction in end-session concentration relative to a counterfactual with ordinary elimination alone over four hours is:

1-\exp[-\widehat{k}_{\mathrm{HD}}T] = 1-\exp[-(0.10667)(4.00)] = 1-\exp(-0.42668) \approx 0.347.

Thus, \(34.7\%\) is the proportional reduction in end-session concentration relative to ordinary elimination alone, not the fraction of the initial amount removed by dialysis. The fraction of the initial amount removed through the dialysis-specific process while both elimination processes act is:

\frac{\widehat{k}_{\mathrm{HD}}}{k_{\mathrm{el}}+\widehat{k}_{\mathrm{HD}}} \left[1-\exp[-(k_{\mathrm{el}}+\widehat{k}_{\mathrm{HD}})T]\right] = \frac{0.10667}{0.100+0.10667} \left[1-\exp[-(0.100+0.10667)(4.00)]\right] \approx 0.290.

For these inputs, the model estimates about \(29.0\%\) of the initial central-compartment amount was removed through the dialysis-specific process. This is a model-based estimate, not a direct measurement of the total drug mass recovered in dialysate.

From an Elimination Rate to Clearance

If an appropriate apparent distribution volume \(V\) is available and the one-compartment assumptions are reasonable, the dialysis-specific clearance is \(CL_{\mathrm{HD}}=k_{\mathrm{HD}}V\). For a concentration in \(\mathrm{mg/L}\), \(V\) in L, and a rate in \(\mathrm{h}^{-1}\), clearance is in \(\mathrm{L/h}\). The volume must be defined consistently with the model; using an unrelated population-average volume can make the clearance estimate misleading.

Worked Example: Converting the Rate Estimate to Clearance

Continue the hypothetical example above and suppose the relevant one-compartment volume is \(40.0\ \mathrm{L}\). Then:

\widehat{CL}_{\mathrm{HD}} = \widehat{k}_{\mathrm{HD}}V = (0.10667\ \mathrm{h}^{-1})(40.0\ \mathrm{L}) = 4.267\ \mathrm{L/h}.

This value is the additional clearance represented by the simple model during dialysis. It is not total body clearance during the session: the model also includes ordinary elimination, with corresponding clearance \(k_{\mathrm{el}}V=(0.100)(40.0)=4.00\ \mathrm{L/h}\). The estimated combined clearance while dialysis is operating is therefore \(8.267\ \mathrm{L/h}\), provided both clearances act on the same modelled compartment.

Sampling Time, Rebound, and Model Diagnostics

The end-of-session sample needs an operational definition. A concentration drawn immediately at the end of treatment may differ from one drawn after blood has redistributed between central and peripheral compartments. Post-dialysis rebound can raise the measured concentration even when no drug has entered the body. Treating a rebound sample as if it were the immediate end-of-session value makes the calculated decline look smaller and therefore tends to underestimate the dialysis-related rate.

Worked Example: How Rebound Changes an Apparent Estimate

Suppose the hypothetical \(8.00\ \mathrm{mg/L}\) pre-dialysis value is followed by an immediate end-of-session value of \(3.50\ \mathrm{mg/L}\), then a value of \(4.20\ \mathrm{mg/L}\) one hour later. The rise is:

\frac{4.20-3.50}{3.50}\times 100\% = 20.0\%.

If the \(4.20\ \mathrm{mg/L}\) rebound sample is incorrectly substituted as the session-end value, the apparent additional rate becomes:

\frac{\ln(8.00/4.20)}{4.00}-0.100 = \frac{0.64436}{4.00}-0.100 = 0.06109\ \mathrm{h}^{-1}.

This is substantially below the \(0.10667\ \mathrm{h}^{-1}\) estimate using the immediate end-of-session sample. The calculation has folded redistribution into the apparent dialysis effect. A simple correction is not justified from these three concentrations alone; a richer distribution model or a clearly specified sampling model may be needed.

Useful diagnostics include plotting every concentration against its actual time, marking the start and end of dialysis and all doses, and checking whether the decline during dialysis is approximately log-linear. A semilog plot that bends systematically, a negative estimated \(k_{\mathrm{HD}}\), large variation between sessions, or a clear post-session rise are warnings against interpreting one constant dialysis rate literally. Assay limits, sample handling, access to the sampling line, and whether samples were taken before or after a dose should also be reviewed.

Comparing Supplementation Strategies by Simulation

Once a concentration profile has been specified, candidate schedules can be compared by simulating concentration over a chosen time window. The simplest illustration treats a supplement as an instantaneous concentration increment \(\Delta C\), equal to dose divided by the effective volume. This approximation makes the timing effect transparent, but it is not a substitute for modelling absorption and distribution when the real supplement is oral or has delayed input.

For a post-dialysis starting concentration \(C_0\), ordinary elimination rate \(k_{\mathrm{el}}\), and a single supplement at time \(\tau\), the profile after the supplement is \(C(t)=C_0e^{-k_{\mathrm{el}}t}+\Delta C e^{-k_{\mathrm{el}}(t-\tau)}\) for \(t\geq\tau\). Before \(\tau\), only the first term applies. The AUC over a finite window \(0\) to \(H\), when \(\tau\leq H\), is:

\mathrm{AUC}_{0-H} = \frac{C_0(1-e^{-k_{\mathrm{el}}H})}{k_{\mathrm{el}}} + \frac{\Delta C(1-e^{-k_{\mathrm{el}}(H-\tau)})}{k_{\mathrm{el}}}.

The second term follows by integrating the supplement’s contribution from its administration time \(\tau\) to the end of the window \(H\). Delaying an otherwise identical supplement shortens the time it contributes to that window’s AUC. It may nevertheless produce a higher concentration late in the window, so total AUC alone cannot determine which schedule is preferable. The relevant exposure target and safety constraints must be drug- and context-specific.

Worked Example: Immediate Versus Delayed Supplementation

Consider a hypothetical post-dialysis concentration \(C_0=5.392\ \mathrm{mg/L}\), an ordinary elimination rate \(k_{\mathrm{el}}=0.10024\ \mathrm{h}^{-1}\), and a supplement represented by an instantaneous \(2.00\ \mathrm{mg/L}\) increment. Compare giving it immediately at hour 0 with giving it at hour 4. No further doses or dialysis occur during the next 24 hours.

With no supplement, the concentration at hour 4 is:

5.392e^{-(0.10024)(4)} = 5.392e^{-0.40096} \approx 3.611\ \mathrm{mg/L}.

Under the delayed strategy, the concentration immediately after the hour-4 supplement is \(3.610898+2.00=5.610898\ \mathrm{mg/L}\), or approximately \(5.611\ \mathrm{mg/L}\). The immediate strategy instead has concentration \((5.392+2.00)e^{-(0.10024)(4)}\approx4.950\ \mathrm{mg/L}\) at hour 4. Thus the delayed strategy is higher at that particular time.

For the 24-hour AUC, the baseline contribution is:

\frac{5.392(1-e^{-(0.10024)(24)})}{0.10024} \approx 48.94\ \mathrm{mg\cdot h/L}.

The immediate supplement contributes \(2.00(1-e^{-2.40576})/0.10024\approx18.15\ \mathrm{mg\cdot h/L}\), giving a total of about \(67.09\ \mathrm{mg\cdot h/L}\). The delayed supplement contributes \(2.00(1-e^{-2.0048})/0.10024\approx17.26\ \mathrm{mg\cdot h/L}\), giving about \(66.20\ \mathrm{mg\cdot h/L}\). The immediate strategy has the larger AUC within this fixed window because the supplement is present for longer; the delayed strategy has the higher concentration at hour 4 and also retains more of its contribution late in the window. These results describe this specified simulation, not a general clinical preference.

Choosing a Model and Avoiding Overinterpretation

The one-compartment calculation is attractive because it is transparent and can be checked by hand. Its limitations matter. Dialysis clearance can vary with blood and dialysate flow, membrane, session duration, access performance, and drug binding. Distribution into peripheral tissues can produce rebound. A dose given during or soon after treatment adds input that must be modelled explicitly. For oral supplements, absorption rate and bioavailability affect both the concentration-time curve and AUC.

Alternatives answer different questions. A dialysate mass-balance approach estimates drug recovered in dialysate, but requires reliable collection and accounting for losses. A multi-compartment PK model can represent redistribution and post-session rebound but needs sufficiently informative sampling. A population PK model can represent patient and session variability and can support simulation across a target population, but its assumptions and parameter uncertainty still need assessment. Comparing regimens by simulated target attainment is useful when a justified exposure target is available; a single deterministic profile should not be mistaken for the probability that an individual will meet that target.

For a reproducible analysis, retain the raw dosing and sampling times, document how “pre-dialysis” and “post-dialysis” were defined, record the dialysis prescription, and make the model and simulation inputs explicit. Test sensitivity to plausible values of \(k_{\mathrm{el}}\), dialysis duration, post-session rebound, and dose-input assumptions. When uncertainty is material, simulate it rather than presenting a single clearance or exposure estimate as exact.

Common pitfall: “The concentration dropped by 56%, so dialysis removed 56% of the drug.” This confuses the observed total decline with removal attributable specifically to dialysis. Under the stated model, ordinary elimination also acts during the session; estimate its contribution separately and use correctly timed samples. A post-dialysis rebound sample should not be substituted for an immediate end-of-session value without modelling redistribution.

Check Your Understanding

Work through the questions, then check the solutions below.

A concentration falls from \(10.0\ \mathrm{mg/L}\) to \(5.00\ \mathrm{mg/L}\) during a three-hour session. If \(k_{\mathrm{el}}=0.0800\ \mathrm{h}^{-1}\), calculate \(\widehat{k}_{\mathrm{HD}}\).
For question 1, calculate the modelled dialysis-specific fraction removed over the three-hour session.
Why can a sample collected one hour after dialysis overstate the immediate end-of-session concentration?
With an equal instantaneous supplement, why can a delayed strategy have lower AUC over a fixed window but a higher concentration near the end of that window?

Solutions. (1) \(\ln(10.0/5.00)/3.00-0.0800=0.23105-0.0800=0.1511\ \mathrm{h}^{-1}\). (2) \(\frac{0.15105}{0.23105}(1-e^{-(0.23105)(3.00)})\approx0.327\), or \(32.7\%\). (3) Redistribution from peripheral compartments can raise the measured central concentration after treatment, even without a dose; this rebound means the later value is not the immediate session-end value. (4) The delayed supplement contributes for less time to the finite-window integral, lowering AUC, but has had less time to be eliminated by the end of the window, so its late concentration can be higher.