Deconvolution Using an Intravenous Disposition Reference

Why Use an Intravenous Disposition Reference?

After an extravascular dose, the plasma concentration reflects both when drug enters the systemic circulation and how the body distributes and eliminates it. The earlier material on connecting dissolution to absorption through convolution describes this relationship: concentration is the accumulated response to drug entering over time. Deconvolution works in the reverse direction. Given a concentration profile and a suitable disposition response, it estimates the rate at which drug appeared systemically.

An intravenous bolus can provide that response because its dose enters the systemic circulation at a known time. The resulting concentration-time profile, divided by the administered dose, is a dose-normalized disposition function. In this tutorial, the estimated input is the rate of systemic drug appearance. For an oral product, that rate reflects the effects of absorption and bioavailability; it is not automatically the rate of drug release from the dosage form or the rate at which drug crosses the gut wall.

For an intravenous bolus of dose \(D_{\mathrm{IV}}\), the reference estimate is \(h(t)=C_{\mathrm{IV}}(t)/D_{\mathrm{IV}}\), after appropriate accounting for baseline concentration and any relevant sample or dose corrections. If the disposition is represented by a one-compartment model with volume \(V\) and elimination rate constant \(k\), then \(h(t)=e^{-kt}/V\). An empirical measured reference can also be used; it need not be forced into a one-compartment form.

Assumptions and Reference Selection

Deconvolution does not create information about the disposition process. Its result is meaningful only if the reference describes the disposition that would apply to the drug entering after the test dose. The central assumptions are:

• Linearity: the concentration response to multiple input amounts is the sum of their individual responses. Clearance, distribution, and other relevant disposition properties must not change materially with concentration or dose.
• Time invariance: the response to an input at a given time is the same as the response to the same input at another time, apart from the shift in time. There must not be a material change in disposition during the observation period.
• Reference compatibility: the IV and extravascular observations represent the same active drug and sufficiently comparable physiological and analytical conditions. A change in clearance or distribution between occasions can be mistaken for a change in input.
• Suitable reference data: the IV profile must adequately characterize the response over the times needed for deconvolution. Dose normalization is valid only when the IV disposition is dose proportional.
• Adequate sampling and measurement: samples must resolve meaningful changes in the concentration profile, and assay error, baseline effects, and timing errors must not dominate the signal.

A useful reference is usually obtained from the same drug, with a known IV dose and sampling that captures both early distribution and later disposition. A reference from a different occasion or population may still be usable, but the analyst should justify comparability rather than assume it. If the IV dose is infused over time rather than given as a bolus, its observed concentration profile already reflects a non-instantaneous input; it cannot simply be divided by dose and treated as a bolus response without accounting for the infusion.

From Convolution to a Discrete Input Profile

Concentration samples are discrete, so a practical calculation represents the input as constant within each sampling interval. Suppose samples are taken at \(t_j=j\Delta\), and let \(R_k\) be the input rate during \(((k-1)\Delta,k\Delta]\). The contribution at time \(t_j\) from that interval depends on the response integrated over the corresponding range of elapsed times. Define

Then the discrete convolution is \(C_j=\sum_{k=1}^{j}R_k w_{j-k}\). The first observation contains only the first interval's input. At each subsequent observation, the current interval contributes through \(w_0\), while earlier intervals contribute through the appropriate later weights.

This recurrence is the simplest discrete deconvolution. Its logic follows directly from isolating the term \(R_jw_0\) in the convolution sum. It also reveals an important limitation: each new rate is obtained by subtracting the predicted contributions of earlier rates, then dividing by \(w_0\). If \(w_0\) is small, errors in concentrations, previous rates, or weights can be magnified.

For a one-compartment disposition function \(h(t)=e^{-kt}/V\), the weights can be integrated exactly:

Numerical Sensitivity and Diagnostics

The recurrence is algebraically straightforward but may be numerically fragile. Concentration measurement error at one time affects the rate recovered at that time and can also affect later rates, because later calculations subtract contributions from all previously estimated intervals. A small \(w_0\) magnifies these errors. Coarse sampling can also conceal rapid input changes, while very late samples may contain little signal relative to assay noise.

Negative recovered rates are a warning, not evidence that drug entered the body in a physically negative amount. They can arise when noisy or rounded concentrations are inconsistent with the assumed response, when the IV reference is mismatched, or when the interval-based approximation is inadequate. A negative rate should prompt examination of the data, units, sampling times, reference function, and assumptions. Silently replacing it with zero changes the estimated profile and should not be presented as a neutral correction.

Useful diagnostics include plotting observed concentrations alongside concentrations reconstructed by reconvolving the estimated input with \(h(t)\), inspecting the recovered rates for abrupt oscillation or implausible values, and checking sensitivity to sampling interval and reference estimation. Agreement after reconvolution is necessary for numerical consistency, but not sufficient evidence of a correct input profile: different input profiles can be difficult to distinguish when data are sparse or noisy.

Alternatives, Limitations, and Common Mistakes

The recurrence above is a direct, interval-based numerical method. It is transparent and easy to audit, but can be sensitive to measurement error. A parametric compartment model instead specifies a smooth input function and estimates its parameters by fitting predicted concentrations. This can stabilize estimation when the chosen model is appropriate, but it restricts the input to the assumed functional form. Smoothing or regularized deconvolution can reduce erratic estimates by penalizing rapid changes, but the penalty introduces a modeling choice and may smooth away real features. These approaches answer related but not identical questions; none removes the need for a suitable disposition reference.

Deconvolution is also limited by the available sampling window. Input occurring before the first sample, or input after the last informative sample, may be poorly estimated. A long disposition tail can carry contributions from earlier input into later concentrations, so truncating the reference too early can distort the recovered profile. For oral dosing, the estimated systemic input combines absorption with the fraction reaching systemic circulation; deconvolution alone does not separate those processes.

• Mistake: treating the measured IV concentration curve as \(h(t)\) without dividing by the IV dose. Correction: normalize the IV response to concentration per unit amount, and confirm dose proportionality.
• Mistake: interpreting every jagged or negative estimate as a real feature of absorption. Correction: check noise amplification, interval width, reference compatibility, and model assumptions before assigning physiological meaning.
• Mistake: assuming that a concentration curve that can be reconstructed proves the recovered input is unique or correct. Correction: reconstruction checks consistency with the chosen response; sparse or noisy data can still leave substantial uncertainty about the input profile.