From standard textbooks · RGUHS Practical Notes
Definitions, parameters, core equations, graphical analysis and TDM applications
Assumptions
k = 0.693/t½ = ln(Cp1/Cp2)/(t2−t1). From semilog slope: k = −slope × 2.303. Units: h⁻¹ S&Y Ch.4t½ = 0.693/k = (0.693 × Vd)/Cl. After 4 t½: ~94% eliminated; after 5 t½: ~97%. Time to steady state = 4–5 t½. DiPiro L3Vd = Dose/Cp⁰. NOT a real anatomical space — reflects partitioning between plasma and tissues. S&Y Ch.4Cl = k × Vd = Dose/AUC₀→∞. Total Cl = ClR + ClH + ClOther. S&Y Ch.4AUC₀→∞ = Cp⁰/k = Dose/Cl. Units: mg·h/L. S&Y Ch.4Vd Reference Table
| Vd Range | Body Compartment | % Body Weight | Drug Examples |
|---|---|---|---|
| ~0.04–0.06 L/kg | Plasma only | ~4–6% | Heparin, warfarin (highly protein-bound) |
| ~0.1–0.3 L/kg | Extracellular fluid | ~10–30% | Aminoglycosides (gentamicin ~0.25 L/kg) |
| ~0.5–0.7 L/kg | Total body water | ~50–70% | Theophylline, ethanol |
| >1 L/kg | Extensive tissue binding | >100% | Digoxin (7 L/kg), chloroquine, TCAs |
| Parameter | Equation | Notes |
|---|---|---|
| Plasma conc. (exponential) | Cp = Cp⁰ × e^(−kt) | Monoexponential decay |
| Plasma conc. (logarithmic) | log Cp = log Cp⁰ − (k/2.303)×t | Straight line on semilog; slope = −k/2.303 |
| Initial concentration | Cp⁰ = Dose / Vd | Extrapolated from y-intercept |
| Elimination rate constant | k = −slope × 2.303 | From semilog plot |
| Half-life | t½ = 0.693 / k | = (0.693 × Vd) / Cl |
| Volume of distribution | Vd = Dose / Cp⁰ | Units: L or L/kg |
| Total clearance | Cl = k × Vd | = Dose / AUC₀→∞ |
| AUC (total) | AUC₀→∞ = Cp⁰ / k | = Dose/Cl; units: mg·h/L |
| AUC (trapezoidal) | AUC₀→t = Σ[(Cn+Cn+1)/2 × Δt] | Sum of trapezoids |
| AUC (terminal) | AUCt→∞ = Cp(last) / k | Added to trapezoid AUC |
| Parameter | Equation | Notes |
|---|---|---|
| Cp during infusion | Cp = (R₀/Cl)(1 − e^(−kt)) | R₀ = infusion rate (mg/h) |
| Steady-state conc. | Css = R₀ / Cl = R₀ / (k × Vd) | Reached after 4–5 t½ |
| Post-infusion decline | Cp = Css × e^(−k×(t−T)) | T = total infusion duration |
| Loading dose | DL = Css × Vd | IV bolus at infusion start |
| Parameter | Equation | Notes |
|---|---|---|
| Plasma conc. (oral) | Cp = (F×D×ka)/[Vd(ka−k)] × (e^(−kt) − e^(−kat)) | F = bioavailability; ka = absorption rate constant |
| Tmax | Tmax = ln(ka/k) / (ka − k) | Time of peak plasma concentration |
| Bioavailability | F = AUC(oral) / AUC(IV) | Dose-normalise if doses differ |
| Flip-flop kinetics | Terminal slope = −ka/2.303 | When ka << k; depot formulations |
Effect of disease on PK DiPiro L11
Given — plasma concentrations after 80 mg IV bolus
Aim, requirements, 10-step method, observation table, results table, graph record, viva questions
To determine the pharmacokinetic parameters — elimination rate constant (k), half-life (t½), apparent volume of distribution (Vd), total clearance (Cl), and area under the curve (AUC) — of a drug following one-compartment model kinetics after intravenous bolus administration.
Administer the IV bolus dose at t = 0. Record exact dose, time, and patient weight.
Collect blood samples at pre-specified time intervals. Label each sample with exact collection time.
Centrifuge at 3000 rpm for 10 min; separate plasma. Store at −20°C until assay.
Measure plasma drug concentrations (Cp) by validated assay. Record in observation table.
Calculate ln Cp and log Cp for each time point. Compute AUC segments using trapezoidal rule.
Plot log Cp (y-axis, logarithmic scale) vs time (x-axis, linear) on semilog paper. Single straight line confirms one-compartment model.
Draw best-fit line. Read Cp⁰ from y-intercept (t = 0 extrapolation).
Calculate slope from two points on best-fit line: Slope = (log C2 − log C1)/(t2 − t1). k = −slope × 2.303.
Calculate all PK parameters using equations in the results table.
State clinical interpretation: Vd distribution compartment, t½ relevance to dosing interval, Cl vs normal values.
| Time (h) | Cp (mg/L) | ln Cp | log Cp | Δt (h) | AUC segment (mg·h/L) |
|---|---|---|---|---|---|
| 0.5 | — | — | — | 0.5 | — |
| 1 | — | — | — | 0.5 | — |
| 2 | — | — | — | 1 | — |
| 4 | — | — | — | 2 | — |
| 6 | — | — | — | 2 | — |
| 8 | — | — | — | 2 | — |
| 12 | — | — | — | 4 | — |
AUC₀→t (total trapezoidal) = Σ all AUC segments = ________ mg·h/L
| Parameter | Formula | Calculated value | Units |
|---|---|---|---|
| Cp⁰ (y-intercept) | Dose / Vd | _______ | mg/L |
| k (elimination rate constant) | −slope × 2.303 | _______ | h⁻¹ |
| t½ (half-life) | 0.693 / k | _______ | h |
| Vd (volume of distribution) | Dose / Cp⁰ | _______ | L or L/kg |
| Cl (total clearance) | k × Vd | _______ | L/h |
| AUC₀→t (trapezoidal) | Σ [(Cp1+Cp2)/2 × Δt] | _______ | mg·h/L |
| AUCt→∞ (terminal) | Cp(last) / k | _______ | mg·h/L |
| AUC₀→∞ (total) | AUC₀→t + AUCt→∞ | _______ | mg·h/L |
Paste or draw your semilog plot here
Mark Cp⁰, slope calculation points, and best-fit line.
State whether data confirm one-compartment or two-compartment model.
Graph: log Cp vs Time (h) — Semilog Plot
5 simple + 5 moderate — full worked solutions with numbered steps
Simple Problems — P01 to P05
Given
Find: k and t½
Given
Find: Vd; state body compartment
Given
Find: Cl and AUC₀→∞ (two methods)
Given
Find: Css and time to reach Css
Given
Find: Loading dose (DL) and maintenance infusion rate (R₀)
Moderate Problems — P06 to P10
Given — 100 mg IV bolus
| t (h) | 1 | 2 | 4 | 6 | 8 | 10 |
|---|---|---|---|---|---|---|
| Cp (mg/L) | 8.6 | 7.0 | 4.6 | 3.0 | 2.0 | 1.3 |
Find: k, t½, Cp⁰, Vd, Cl, AUC₀→∞
Given
Find: Tmax and Cmax
Given
Find: Absolute bioavailability (F)
Given
Find: Total Vd, DL, maintenance dose, expected trough
Given
Find: Adjusted k, t½; new dosing regimen to maintain same peak
60+ formulae across 12 categories — searchable and filterable
| Parameter / Formula | Equation | Notes & Source |
|---|---|---|
| 1. First-Order Elimination Kinetics | ||
| Plasma conc. at time t | Cp(t) = Cp⁰ × e^(−kt) | Monoexponential decay; one-compartment IV bolus S&Y Ch.4 |
| Plasma conc. (semilog form) | log Cp = log Cp⁰ − (k/2.303)×t | Straight line on semilog paper; slope = −k/2.303 S&Y Ch.4 |
| k from semilog slope | k = −slope × 2.303 | Slope is negative; k is always positive; units: h⁻¹ S&Y Ch.4 |
| k from two time points | k = ln(Cp1/Cp2) / (t2 − t1) | t2 > t1; equivalent to slope×2.303 method DiPiro L3 |
| Half-life | t½ = 0.693 / k | Independent of dose for first-order kinetics S&Y Ch.4 |
| Half-life (extended form) | t½ = (0.693 × Vd) / Cl | Explains why t½ changes with disease altering Vd or Cl DiPiro L3 |
| % drug remaining at time t | % = e^(−kt) × 100 | After 1 t½: 50%; 2 t½: 25%; 4 t½: 6.25% DiPiro L3 |
| % drug eliminated | % = (1 − e^(−kt)) × 100 | 4 t½: 93.75%; 5 t½: 96.9% DiPiro L3 |
| Amount after n half-lives | A(n) = A₀ × (0.5)^n | n = number of half-lives elapsed S&Y Ch.2 |
| 2. Volume of Distribution (Vd) | ||
| Vd (IV bolus) | Vd = Dose / Cp⁰ | Cp⁰ = initial conc. extrapolated to t=0 S&Y Ch.4 |
| Vd (from AUC) | Vd = Dose / (k × AUC₀→∞) | Model-independent; does not require Cp⁰ S&Y Ch.4 |
| Vd (protein binding effect) | Vd = Vp + Vt×(fu/fut) | fu = plasma unbound fraction; fut = tissue unbound; Vp≈3 L S&Y Ch.9 |
| Vd (steady state, 2-cpt) | Vdss = Vc + Vt | Vc = central; Vt = peripheral; Vdss unaffected by kinetics S&Y Ch.5 |
| 3. Clearance (Cl) | ||
| Total clearance | Cl = k × Vd = Dose / AUC₀→∞ | Model-independent second form S&Y Ch.4 |
| Cl relationship | Cl = ClR + ClH + ClOther | Sum of all routes DiPiro L3 |
| Renal clearance | ClR = Cl × fe | fe = fraction excreted unchanged in urine DiPiro L11 |
| Hepatic clearance | ClH = Q × ER | Q = hepatic blood flow (~90 L/h); ER = extraction ratio DiPiro L9 |
| High ER drugs (flow-limited) | ClH ≈ Q (hepatic blood flow) | ER > 0.7; sensitive to changes in Q (e.g. propranolol) DiPiro L9 |
| Low ER drugs (capacity-limited) | ClH = fu × CLint | ER < 0.3; sensitive to protein binding (e.g. warfarin) DiPiro L9 |
| Creatinine clearance (C-G) | ClCr = [(140−age)×wt] / (72×SCr) | Multiply by 0.85 for females; wt in kg; SCr in mg/dL DiPiro L11 |
| 4. Area Under the Curve (AUC) | ||
| AUC₀→∞ (IV bolus) | AUC = Cp⁰ / k = Dose / Cl | Total drug exposure from t=0 to infinity S&Y Ch.4 |
| AUC (linear trapezoidal) | AUC₀→t = Σ [(Cn+Cn+1)/2 × Δt] | Linear trapezoidal; use for ascending or flat sections DiPiro L3 |
| AUC (log-linear trapezoidal) | AUC = (Cn−Cn+1)/ln(Cn/Cn+1) × Δt | Better for declining log-linear segments S&Y Ch.4 |
| AUC (terminal extrapolation) | AUCt→∞ = Cp(last) / k | Added to trapezoid AUC for total AUC₀→∞ DiPiro L3 |
| AUC (oral dose) | AUC = F×Dose / Cl | F = bioavailability; accounts for incomplete absorption S&Y Ch.8 |
| AUMC (1st moment) | AUMC₀→∞ = Cp⁰ / k² | For non-compartmental analysis; used to calc MRT S&Y Ch.9 |
| Mean Residence Time | MRT = AUMC / AUC = 1/k | 1/k for one-compartment IV bolus; 1/k + τ/2 for infusion S&Y Ch.9 |
| 5. IV Bolus — One-Compartment | ||
| Cp at time t | Cp = Cp⁰ × e^(−kt) | Most fundamental equation S&Y Ch.4 |
| Initial concentration | Cp⁰ = Dose / Vd | Concentration if distribution were truly instantaneous S&Y Ch.4 |
| Cp (log form) | log Cp = log Cp⁰ − (k/2.303)t | For semilog plot; y-intercept = log Cp⁰ S&Y Ch.4 |
| 6. IV Infusion — One-Compartment | ||
| Cp during infusion (t < T) | Cp = (R₀/Cl)(1 − e^(−kt)) | R₀ = infusion rate (mg/h); approaches Css asymptotically S&Y Ch.6 |
| Steady-state concentration | Css = R₀ / Cl = R₀ / (k×Vd) | Reached after 4–5 t½ regardless of R₀ S&Y Ch.6 |
| % of Css at time t | % = (1 − e^(−kt)) × 100 | At 1 t½: 50%; 3.32 t½: 90%; 4 t½: 93.75% DiPiro L4 |
| Post-infusion decline (t > T) | Cp = Css × e^(−k(t−T)) | T = total infusion duration S&Y Ch.6 |
| Loading dose | DL = Css × Vd | Give as IV bolus before starting infusion S&Y Ch.6 |
| Required infusion rate | R₀ = Css(target) × Cl | Design equation for target-driven dosing DiPiro L4 |
| 7. Oral Absorption — One-Compartment | ||
| Cp after oral dose | Cp = (F×D×ka)/[Vd(ka−k)]×(e^−kt−e^−kat) | Biexponential; ka > k for normal absorption S&Y Ch.8 |
| Time of peak (Tmax) | Tmax = ln(ka/k) / (ka − k) | At Tmax, absorption rate = elimination rate S&Y Ch.8 |
| Peak concentration (Cmax) | Cmax = Cp at t = Tmax | Substitute Tmax into Cp(t) equation S&Y Ch.8 |
| Flip-flop kinetics | Terminal slope = −ka/2.303 | When ka << k; absorption is rate-limiting; depot formulations S&Y Ch.8 |
| Lag time (tlag) | Modified Tmax = tlag + ln(ka/k)/(ka−k) | Add tlag to Tmax formula if absorption not immediate S&Y Ch.8 |
| 8. Bioavailability & Bioequivalence | ||
| Absolute bioavailability | F = (AUCoral×DIV) / (AUCIV×Doral) | Dose-normalise if different doses used S&Y Ch.7 |
| Relative bioavailability | Frel = AUCtest / AUCreference | For generic vs innovator (must use same doses) S&Y Ch.7 |
| Hepatic first-pass F | F = 1 − ER | ER = hepatic extraction ratio; high-ER drugs have low F DiPiro L9 |
| Overall oral F | F = fa × fg × fh | fa = fraction absorbed; fg = gut wall; fh = hepatic fraction S&Y Ch.7 |
| Bioequivalence criterion | 90% CI of Cmax, AUC ratio: 80–125% | FDA/EMA standard for generic approval S&Y Ch.7 |
| 9. Multiple Dosing & Steady State | ||
| Css(max) at steady state | Css,max = (F×D/Vd)/(1−e^(−kτ)) | τ = dosing interval; instantaneous absorption assumed S&Y Ch.9 |
| Css(min) at steady state | Css,min = Css,max × e^(−kτ) | Just before the next dose S&Y Ch.9 |
| Average Css | Css(avg) = F×D / (Cl×τ) | = AUCss/τ; fundamental target for dosing design DiPiro L4 |
| Accumulation factor (R) | R = 1 / (1 − e^(−kτ)) | Ratio of Css to single-dose Cp at same time S&Y Ch.9 |
| Time to steady state | 4–5 × t½ | Independent of dose, route, or frequency DiPiro L4 |
| Dosing interval design | τ = ln(Cmax/Cmin) / k | Design equation: choose target Cmax and Cmin DiPiro L4 |
| Loading dose (multiple dosing) | DL = Dm / (1 − e^(−kτ)) | Dm = maintenance dose; gives Css from first dose S&Y Ch.9 |
| 10. Non-linear (Michaelis-Menten) Pharmacokinetics | ||
| Rate of drug elimination | v = Vmax×C / (Km + C) | Vmax = max rate (mg/h); Km = Michaelis constant (mg/L) S&Y Ch.10 |
| High conc. (C >> Km) | v ≈ Vmax (zero-order) | Saturated enzymes; rate is constant regardless of Cp S&Y Ch.10 |
| Low conc. (C << Km) | v ≈ (Vmax/Km)×C (first-order) | Approximates linear kinetics; clearance = Vmax/Km S&Y Ch.10 |
| Phenytoin Css | Css = Km×DR / (Vmax−DR) | Css rises disproportionately with dose rate S&Y Ch.10 |
| Half-life (non-linear) | Not constant; increases with Cp | Cannot use t½ for dosing interval design S&Y Ch.10 |
| AUC (non-linear) | Not proportional to dose | Small dose increase → large AUC increase (phenytoin toxicity risk) S&Y Ch.10 |
| 11. Protein Binding | ||
| Fraction unbound in plasma | fu = Cu / Ctotal | Cu = free conc.; warfarin fu = 0.01 (99% bound) S&Y Ch.9 |
| Fraction bound | % bound = (1 − fu) × 100 | Only unbound drug crosses membranes and exerts effect S&Y Ch.9 |
| Effect of binding on Vd | Vd ↑ as fu ↑ (or fut ↓) | More unbound drug leaves plasma for tissues S&Y Ch.9 |
| Effect on Cl (capacity-limited) | Cl = fu × CLint | Low-ER drugs: Cl proportional to fu; binding affects Cl DiPiro L9 |
| Effect on Cl (flow-limited) | Cl ≈ hepatic blood flow Q | High-ER drugs: protein binding does NOT significantly alter Cl DiPiro L9 |
| 12. Pharmacodynamics (PK/PD) | ||
| Emax model | E = Emax × C / (EC50 + C) | Emax = max effect; EC50 = conc. at 50% effect DiPiro L7 |
| Sigmoid Emax (Hill equation) | E = Emax × C^n / (EC50^n + C^n) | n = Hill coefficient; n>1: steep; n<1: shallow S-curve DiPiro L7 |
| Log-linear model | E = m × log C + b | Valid for 20–80% of effect range; m = slope DiPiro L7 |
| Therapeutic index | TI = TD50 / ED50 | Ratio of toxic to effective median doses; narrow TI needs TDM S&Y |
| Conc.-effect at Css | E = Emax × Css / (EC50 + Css) | At steady state; basis for target concentration approach DiPiro L7 |
| PK/PD target (bactericidal) | AUC/MIC or Cmax/MIC | Aminoglycosides: Cmax/MIC > 8–10; fluoroquinolones: AUC/MIC > 125 DiPiro |
| PK/PD target (time-dependent) | T > MIC (% of dosing interval) | β-lactams: T>MIC > 40–70% for bactericidal activity DiPiro |
From standard textbooks