Educational model, not a clinical tool.
MoreDrugs T1–T4 are fictional teaching drugs with textbook-typical parameters, and named-drug doses are teaching examples. Never use these numbers to choose doses for real patients: follow current national guidelines, your hospital's protocols and your supervisors.
Drug particles in the body
t 0 h C 0 mg/L Status —

Key facts to remember

  • Dose: D/H × V. Convert units first (1 g = 1000 mg = 1 000 000 mcg). Only scored tablets may be halved.
  • Drip rate: drops/min = volume (mL) × drop factor ÷ time (min). Macro-drip 20 drops/mL; micro-drip 60 drops/mL, where drops/min = mL/h.
  • Infusions: mL/h = dose (mcg/kg/min) × weight (kg) × 60 ÷ concentration (mcg/mL).
  • Maintenance fluids (Holliday–Segar): 100 / 50 / 20 mL/kg/day for the first 10 kg / next 10 kg / each kg above 20 (4-2-1 mL/kg/h).
  • Half-life: t½ = 0.693 × Vd / CL. Steady state takes 4–5 half-lives whatever the dose; after stopping, most drug is gone in 4–5 half-lives.
  • Loading dose = Ctarget × Vd / F (set by Vd). Maintenance dose rate = Ctarget × CL / F (set by CL).
  • Renal impairment lowers the clearance of renally excreted drugs (gentamicin, digoxin, magnesium): lengthen the interval or reduce the dose, guided by levels.
  • Killing patterns: aminoglycosides are concentration-dependent (once-daily peak); β-lactams are time-dependent (regular intervals above the MIC).
  • Adherence: missed doses let levels fall below the MEC; for ARVs this allows viral replication and resistance. Never double a missed dose.

ADME: what the body does to a drug

Pharmacokinetics (PK) describes how drug concentration changes with time as the body Absorbs, Distributes, Metabolises and Excretes it. Pharmacodynamics (PD) describes what the drug does to the body at a given concentration. The left panel of the lab shows ADME as moving particles: an oral dose enters the gut, passes the liver (first-pass metabolism) and reaches the blood; drug then distributes into tissues and is cleared by the kidneys (to urine) and liver (metabolites).

One-compartment model, IV bolus

Treat the body as one well-mixed volume, the apparent volume of distribution $V_d$ (L). Elimination is first-order: the rate is proportional to the amount present. With clearance $CL$ (L/h, the volume of plasma completely cleared of drug per unit time):

$$k=\frac{CL}{V_d},\qquad C(t)=\frac{D}{V_d}\,e^{-kt},\qquad t_{1/2}=\frac{\ln 2}{k}\approx\frac{0.693}{k}$$

On a log scale the curve is a straight line of slope $-k$. Each half-life the concentration halves; after 4 half-lives about 94% and after 5 about 97% of a single dose has gone. $V_d$ is "apparent": a drug that binds strongly to tissue can have a $V_d$ of hundreds of litres, far more than body water (about 42 L in a 70 kg adult).

Area under the curve and bioavailability

$$\text{AUC}_{0\to\infty}=\int_0^\infty C\,dt=\frac{F\cdot D}{CL}$$

AUC measures total exposure. Bioavailability $F$ is the fraction of a dose that reaches the systemic circulation unchanged, $F = \text{AUC}_\text{oral}/\text{AUC}_\text{IV}$ for equal doses. Incomplete absorption and first-pass metabolism in gut wall and liver both lower $F$.

Oral dosing: first-order absorption (the Bateman function)

$$C(t)=\frac{F\,D\,k_a}{V_d\,(k_a-k)}\left(e^{-kt}-e^{-k_a t}\right),\qquad t_{\text{max}}=\frac{\ln(k_a/k)}{k_a-k}$$

Faster absorption (larger $k_a$) gives an earlier, higher peak; AUC does not change because it depends only on $F$, $D$ and $CL$.

IV infusion

At a constant rate $R_0$ (mg/h) the concentration rises towards a plateau $C_{ss}=R_0/CL$ as $C(t)=\frac{R_0}{CL}(1-e^{-kt})$. It takes about 3.3 half-lives to reach 90% of the plateau, whatever the rate. The plateau level depends on clearance only; the time to get there depends on half-life only.

Repeated dosing, accumulation and steady state

When a dose $D$ is repeated every $\tau$ hours before the previous dose has been eliminated, drug accumulates until the amount eliminated per interval equals the amount given. For IV bolus doses (one-compartment):

$$R_\text{ac}=\frac{1}{1-e^{-k\tau}},\qquad C_{ss,\max}=\frac{D}{V_d}\cdot\frac{1}{1-e^{-k\tau}},\qquad C_{ss,\min}=C_{ss,\max}\,e^{-k\tau}$$
$$C_{ss,\text{avg}}=\frac{\text{AUC}_\tau}{\tau}=\frac{F\cdot D}{CL\cdot\tau}$$

$C_{ss,\text{avg}}$ depends on the dosing rate $D/\tau$, not on how it is split. Smaller, more frequent doses give the same average with less peak-to-trough swing. The peak-to-trough ratio is $e^{k\tau}$, so to stay inside a window with $\text{MTC}/\text{MEC}=r$ you need $\tau < \ln r / k$.

Loading dose and maintenance dose

$$\text{LD}=\frac{C_\text{target}\cdot V_d}{F},\qquad \text{Maintenance dose rate}=\frac{C_\text{target}\cdot CL}{F}$$

A loading dose fills the volume of distribution at once, so a drug with a long half-life reaches its target immediately rather than after 4–5 half-lives. The loading dose depends on $V_d$; the maintenance dose depends on $CL$.

Two-compartment model

Many drugs first mix in a central compartment (blood and well-perfused organs, $V_1$) and then distribute more slowly into a peripheral compartment (muscle, fat, $V_2$) with inter-compartmental clearance $Q$. After an IV bolus the curve is bi-exponential:

$$C(t)=A\,e^{-\alpha t}+B\,e^{-\beta t},\qquad \alpha+\beta=k_{10}+k_{12}+k_{21},\quad \alpha\beta=k_{10}k_{21}$$

with $k_{10}=CL/V_1$, $k_{12}=Q/V_1$, $k_{21}=Q/V_2$. The steep early α (distribution) phase is followed by the slower β (terminal elimination) phase; the clinically relevant half-life is usually $t_{1/2,\beta}=\ln 2/\beta$.

Organ impairment

Total clearance is the sum of renal and non-renal clearance. In this lab $CL = CL_\text{normal}\,[f_e\cdot(\text{renal function}) + (1-f_e)\cdot(\text{hepatic function})]$, where $f_e$ is the fraction normally excreted unchanged in urine. When clearance falls, $k$ falls, $t_{1/2}$ lengthens, accumulation increases and $C_{ss,\text{avg}}$ rises in proportion. Typical responses are to lower the dose, lengthen the interval, or both, and to monitor levels. (Real dose adjustment uses measured renal function such as creatinine clearance or eGFR and drug-specific guidance.)

The lab solves the linear compartment equations numerically (Runge–Kutta, step 0.01 h) and superposes single-dose responses to find the true steady state, so you can check every formula above against the curve.

Pharmacodynamics: the Emax model and the Hill equation

$$E=\frac{E_{\max}\,C^{\,n}}{EC_{50}^{\,n}+C^{\,n}}$$

$EC_{50}$ is the concentration giving half the maximum effect: a measure of potency (lower $EC_{50}$ = more potent). $E_{\max}$ is efficacy, the largest effect the drug can produce. The Hill coefficient $n$ sets steepness. On a linear axis the curve is a rectangular hyperbola (for $n=1$); on a log axis it becomes the familiar sigmoid, linear between about 20% and 80% of $E_{\max}$.

  • Full agonist: $E_{\max}$ = system maximum. Partial agonist: lower $E_{\max}$ even at full receptor occupancy.
  • Competitive (surmountable) antagonist shifts the curve to the right in parallel. The dose ratio is given by the Schild equation $DR = 1 + [I]/K_i$; $E_{\max}$ is unchanged because enough agonist outcompetes it.
  • Non-competitive (insurmountable) antagonist lowers $E_{\max}$; in the simplest case $E_{\max}' = E_{\max}/(1+[I]/K_i)$ and $EC_{50}$ is unchanged.

Quantal dose–response and therapeutic index

A quantal curve plots the percentage of a population showing an all-or-none response (therapeutic benefit, toxicity) against dose. The therapeutic index is $TI = TD_{50}/ED_{50}$ (or $LD_{50}/ED_{50}$ in animals). Because curves differ in slope, a stricter measure is the certain safety factor $TD_1/ED_{99}$. Drugs with a narrow TI (e.g. digoxin, lithium, aminoglycosides, warfarin) need careful dosing and often therapeutic drug monitoring.

1. Single IV bolus of teaching drug T1

T1: $V_d = 35$ L, $CL = 5$ L/h, dose 200 mg IV.

  1. $k = CL/V_d = 5/35 = 0.1429\ \text{h}^{-1}$
  2. $t_{1/2} = 0.693/0.1429 = 4.85$ h
  3. $C_0 = D/V_d = 200/35 = 5.71$ mg/L
  4. At 6 h: $C = 5.71\,e^{-0.1429\times 6} = 5.71\times 0.424 = 2.42$ mg/L (below the MEC of 4 mg/L)
  5. $\text{AUC}_{0\to\infty} = D/CL = 200/5 = 40$ mg·h/L

Check: set the lab to T1, IV bolus, 1 dose and hover at t = 6 h.

2. Repeated dosing to steady state

Same drug, 200 mg every 6 h.

  1. $e^{-k\tau} = e^{-0.857} = 0.424$, so $R_\text{ac} = 1/(1-0.424) = 1.74$
  2. $C_{ss,\max} = 5.71\times 1.74 = 9.93$ mg/L (below the MTC of 12)
  3. $C_{ss,\min} = 9.93\times 0.424 = 4.21$ mg/L (above the MEC of 4)
  4. $C_{ss,\text{avg}} = D/(CL\,\tau) = 200/(5\times 6) = 6.67$ mg/L
  5. Time to ~90% of steady state $\approx 3.3\,t_{1/2} = 16$ h, i.e. roughly the third or fourth dose.

These are the default lab settings: read the steady-state lines and the stats panel.

3. The same regimen in severe renal impairment

T1 is 90% renally cleared ($f_e = 0.9$). Renal function falls to 20% of normal.

  1. $CL = 5\,[0.9\times 0.2 + 0.1\times 1] = 5\times 0.28 = 1.4$ L/h
  2. $k = 1.4/35 = 0.040\ \text{h}^{-1}$, $t_{1/2} = 17.3$ h (3.6× longer)
  3. Unchanged regimen: $C_{ss,\text{avg}} = 200/(1.4\times 6) = 23.8$ mg/L, far above the MTC. $R_\text{ac}$ rises from 1.74 to 4.69.
  4. Keeping $D/\tau$ proportional to $CL$: 100 mg every 12 h gives $C_{ss,\text{avg}} = 100/(1.4\times 12) = 5.95$ mg/L, with $R_\text{ac} = 1/(1-e^{-0.48}) = 2.62$, $C_{ss,\max} = 2.86\times 2.62 = 7.49$ and $C_{ss,\min} = 7.49\times 0.619 = 4.64$ mg/L: inside the window. Try it.
  5. The same daily dose as 200 mg every 24 h gives the same average, but the swing $e^{0.04\times 24} = 2.61$ is so wide that the trough (3.54 mg/L) falls below the MEC. When the half-life lengthens, how a dose is split still matters.

4. Oral dosing of teaching drug T2

T2: $V_d = 50$ L, $CL = 5$ L/h, $k_a = 1.5\ \text{h}^{-1}$, $F = 0.8$, single 400 mg dose.

  1. $k = 0.10\ \text{h}^{-1}$, $t_{1/2} = 6.93$ h
  2. $t_{\text{max}} = \ln(1.5/0.1)/(1.5-0.1) = 2.708/1.4 = 1.93$ h
  3. $C_{\max} = \frac{0.8\times 400\times 1.5}{50\times 1.4}\left(e^{-0.193}-e^{-2.90}\right) = 6.857\times(0.824-0.055) = 5.27$ mg/L
  4. If repeated every 12 h: $C_{ss,\text{avg}} = 0.8\times 400/(5\times 12) = 5.33$ mg/L

5. Loading dose for a long half-life drug (T3)

T3: $V_d = 300$ L, $CL = 6$ L/h, window 1–2 mg/L, target 1.5 mg/L, given IV every 24 h.

  1. $k = 0.02\ \text{h}^{-1}$, $t_{1/2} = 34.7$ h, so without a loading dose it takes about $3.3\,t_{1/2} = 115$ h, nearly 5 days, to reach 90 % of steady state.
  2. Maintenance: $D = C_\text{target}\,CL\,\tau = 1.5\times 6\times 24 = 216$ mg (use 220 mg)
  3. Simple loading dose: $LD = C_\text{target}\,V_d = 1.5\times 300 = 450$ mg gives 1.5 mg/L at once, but by 24 h it has fallen to $1.5\,e^{-0.48} = 0.93$ mg/L, just below the MEC.
  4. With intermittent boluses, load to the steady-state peak instead: $LD = D\,R_\text{ac} = \dfrac{220}{1-e^{-0.02\times 24}} = \dfrac{220}{0.381} \approx 580$ mg. Then $C_0 = 1.93$ mg/L and the first trough is $1.93\times 0.619 = 1.20$ mg/L: inside the window from the start.

6. Competitive antagonist (Schild)

An agonist has $EC_{50} = 1$ µM. A competitive antagonist is present at $[I] = 4K_i$.

  1. Dose ratio $DR = 1 + 4 = 5$, so the apparent $EC_{50}$ is 5 µM.
  2. $E_{\max}$ is unchanged. With a non-competitive antagonist at the same $[I]/K_i$, $E_{\max}$ would fall to $100/5 = 20\%$.

Each challenge checks the regimen that is currently loaded in the lab. Press Set up to load the drug, then change the regimen (dose, interval, route, loading dose) and press Check.

Guided experiments

  1. Half-life rule. T1, 1 dose IV bolus. Hover the curve at 0, 4.85, 9.7 and 14.6 h. Each value should be half the one before.
  2. Dose rate, not dose size. Compare 200 mg every 6 h with 400 mg every 12 h and 100 mg every 3 h. $C_{ss,\text{avg}}$ stays 6.67 mg/L but the swing changes.
  3. Route matters for the shape, not the AUC. Switch T2 between IV bolus and oral with F = 1. The AUC is identical; the oral peak is lower and later.
  4. Distribution phase. Load T4 (two-compartment) and give one dose. Note the rapid early fall (α phase) and the slower terminal phase (β). Increase Inter-compartmental clearance Q: distribution gets faster. Expect the α phase to steepen and shorten while the terminal β slope barely changes, because elimination (CL) is unchanged.
  5. Renal impairment. With T1 at the default regimen, step renal function from Normal to Severe. Watch the dashed "normal function" curve and the accumulation factor. Record $C_{ss,\text{avg}}$ at each step: expect it to rise in proportion to $1/CL$, with the trough crossing the MTC line, because 90 % of T1 leaves through the kidney.
  6. Potency vs efficacy. In Dose–response mode, drag agonist B's dot left (more potent) and down (less efficacious). Expect a curve that rises at lower doses but plateaus lower: potency is $EC_{50}$, efficacy is $E_{\max}$, and a drug can win on one and lose on the other.

Ward cases

Pharmacokinetics explains rules you will follow on the ward every day. Each case below loads a teaching drug into the concentration–time lab so you can see why the rule exists, then asks one exam-style question. Doses quoted are teaching examples from national and WHO guidance, not prescribing advice.

Teaching examples at a glance

Drug (teaching example)Typical teaching dosePK / practice point
Artemether–lumefantrine (AL) 20/120 mgWeight bands: 5–14 kg 1 tab, 15–24 kg 2, 25–34 kg 3, ≥35 kg 4 tablets per dose; 6 doses at 0, 8, 24, 36, 48, 60 hLumefantrine absorption rises with fat: give with food or milk. Repeat a dose vomited within 30 minutes.
Artesunate IV/IM (severe malaria)3 mg/kg if <20 kg, 2.4 mg/kg if ≥20 kg at 0, 12, 24 h, then daily until oral AL is toleratedVery short half-life; works by rapid parasite killing at high peak levels, not by accumulation. Always complete a full AL course afterwards.
Amoxicillin (pneumonia)40 mg/kg per dose 12-hourly for 5 days (child)Time-dependent killing: what matters is time above the MIC, so doses must be spread through the day.
Gentamicin7.5 mg/kg once daily (child, normal kidneys)Concentration-dependent killing and a post-antibiotic effect: one high daily peak. Renally cleared, so impairment needs a longer interval and levels.
Ceftriaxone50 mg/kg once daily (higher in meningitis)Long half-life (~6–9 h) allows once-daily dosing despite time-dependent killing.
Paracetamol15 mg/kg 6-hourly, maximum 4 doses in 24 hHepatic metabolism; overdose saturates safe pathways and causes liver injury.
Oxytocin10 IU IM after birth (AMTSL); for PPH 20 IU in 1 L normal saline at 60 drops/minHalf-life of minutes: the effect stops quickly when an infusion stops, so infusions must run continuously.
Magnesium sulfate (eclampsia)Loading 4 g IV (20 mL of 20%) over 5–10 min; then 1 g/h IV, or 10 g IM with the loading dose and 5 g IM 4-hourly, for 24 h after delivery or the last fitA loading dose reaches the target level at once instead of after 4–5 half-lives. Before each dose: RR ≥ 16/min, knee reflexes present, urine ≥ 30 mL/h. Antidote: calcium gluconate 1 g (10 mL of 10%) IV slowly. Toxicity rises in renal impairment.
Normal saline 0.9% / Ringer's lactatePlan C: 100 mL/kg (30 mL/kg then 70 mL/kg; 1 h + 5 h in infants, 30 min + 2½ h at 1–5 years)Isotonic fluids stay in the extracellular space; dextrose 5% alone distributes into all body water.

Sources checked: Kenya Ministry of Health Basic Paediatric Protocols (5th ed., 2022); Kenya National Guidelines for the Diagnosis, Treatment and Prevention of Malaria; WHO Guidelines for Malaria (2023) and WHO Pocket Book of Hospital Care for Children; WHO Managing Complications in Pregnancy and Childbirth. Follow the current editions used in your institution.

TermMeaning
VdApparent volume of distribution: amount in body ÷ plasma concentration (L).
CLClearance: volume of plasma cleared of drug per unit time (L/h). Renal + hepatic + other.
kElimination rate constant = CL/Vd (h⁻¹).
t½Time for concentration to halve = 0.693/k.
FBioavailability: fraction of dose reaching systemic circulation.
kaFirst-order absorption rate constant (h⁻¹).
MEC / MTCMinimum effective / minimum toxic concentration; between them lies the therapeutic window.
CssSteady-state concentration (peak, trough or average) when intake equals elimination over each interval.
AUCArea under the concentration–time curve: total exposure (mg·h/L).
EC50Concentration producing 50% of the drug's maximum effect (potency).
EmaxMaximum effect a drug can produce (efficacy).
TITherapeutic index = TD50/ED50.
feFraction of drug excreted unchanged in urine.