Drug Dosing Calculator: One- and Two-Compartment Pharmacokinetics

The model follows linear mass balances. For the central and peripheral amounts, $\dot X_c=I-(k_{10}+k_{12})X_c+k_{21}X_p$ and $\dot X_p=k_{12}X_c-k_{21}X_p$, with $C=X_c/V_c$, $k_{10}=CL/V_c$, $k_{12}=Q/V_c$ and $k_{21}=Q/V_p$. In one-compartment mode, the exchange terms vanish. Oral input has a depot with $\dot X_g=-k_aX_g$ and $I=k_aX_g$; each oral dose adds $FD$ to this available-dose depot.

Bourne, Two Compartment Body Model (2014), PDF pp. 3–4, gives the eigenvalue relations $\alpha+\beta=k_{10}+k_{12}+k_{21}$ and $\alpha\beta=k_{10}k_{21}$. The IV response is $D/V_c\,[w_ae^{-\alpha t}+w_be^{-\beta t}]$, where $w_a=(\alpha-k_{21})/(\alpha-\beta)$ and $w_b=1-w_a$. One-compartment mode uses one exponential with rate $CL/V$. Oral responses convolve each mode with first-order absorption. Equal rates use the continuous limit $k_at e^{-k_at}$; a small-argument exponential series avoids cancellation.

Responses are summed for each administered dose. At steady state, geometric sums replace the finite sums (Bourne, Multiple IV Bolus Dose Administration, 2014, PDF pp. 5–8; Introducing Mathematical Biology, repeated oral doses). A logarithmic time scan brackets each peak before bounded minimization refines it. Trough means the concentration immediately before the next dose. The first-interval maximum is searched only between zero and the next scheduled dose.

Conservation gives $AUC_{ss,\tau}=FD/CL$ and $C_{avg,ss}=FD/(CL\tau)$, with $F=1$ for IV boluses. The clearance multiplier applies for the entire regimen. The observed terminal half-life is $\ln(2)/\min(k_a,\beta)$ for oral dosing and $\ln(2)/\beta$ for IV dosing. Chart intervals include both sides of each IV jump.

Equal positive doses begin at time zero, with no pre-existing drug and no loading dose. Parameters remain constant; kinetics are linear and time invariant. Each disposition compartment is well mixed. Distribution is reversible through a common intercompartmental clearance; elimination occurs only centrally. Bioavailability and first-order absorption are constant, with no lag.

Defaults are illustrative: central volume 40 L, clearance 3 L/h and absorption rate 1.2 h⁻¹ follow Table 4 of WiredWhite post 44027; the 200 mg amount, 12 h interval, 60 L peripheral volume and 8 L/h distribution clearance are author-selected scenarios. No default is a drug-specific recommendation. The comparison lines of 1 and 8 mg/L are illustrative values used in that article.

The schematic is conceptual; its sizes do not represent anatomical volumes. Its small concentration curve is normalized to the plotted scale. Playback changes only the cursor, concentration readout and diagram.

For educational and model-development comparisons of fixed-parameter, one- or two-compartment linear PK with equal oral doses or instantaneous IV boluses. No prescription optimization or clinical safety classification is performed. A comparison band does not establish therapeutic efficacy or toxicity. Final-interval values are not automatically steady-state values.

The terminal disposition half-life differs from the central elimination half-life in a two-compartment model. Slow oral absorption can determine the observable terminal tail (flip-flop kinetics). Apparent parameters (CL/F, V/F and Q/F) must be used consistently with F = 1.

The manual clearance scenario is inspired by the sensitivity questions in WiredWhite post 48377; its multi-drug inhibition, binding and pharmacodynamic models are outside this tool. There is no patient variability, uncertainty interval, parameter estimation, lag, infusion, nonlinear elimination, changing clearance or pharmacodynamic prediction. Use a validated drug-specific method for clinical interpretation.

Source consistency: WiredWhite post 44027, Results and Discussion, describes “peaks exceeding the maximum safe concentration (Cmax)”. Its Tables 1 and 4 specify six 5 mg doses, V = 40 L and a 1–8 mg/L comparison band. Even retaining all six doses centrally would give only 30/40 = 0.75 mg/L, so that narrative cannot follow the listed regimen. This tool uses the stated PK parameters as illustrations and does not treat those plots or concentration claims as validation.