What is the therapeutic index formula?

The therapeutic index (TI) is TD50 divided by ED50: the dose that causes a defined toxic effect in 50% of a population divided by the dose that produces the desired effect in 50%. In animal studies, LD50, the dose lethal to 50% of the animals, replaces TD50. Because it is a ratio of two doses, TI has no units. A higher TI means the toxic and effective doses are far apart, so the drug is safer.

What is the difference between therapeutic index and therapeutic window?

The therapeutic index is a ratio of doses, TD50 divided by ED50, usually derived from population studies. The therapeutic window is a range of blood concentrations, from the minimum effective concentration to the minimum toxic concentration, and it is what clinicians use when they check drug levels. Both describe how much room there is between benefit and harm. Phenytoin's usual window, for example, is a total serum level of about 10 to 20 mcg/mL.

Is a high therapeutic index good or bad?

A high therapeutic index is good: it means the toxic dose is many times larger than the effective dose, so dosing errors or differences between patients are less likely to cause harm. A low, or narrow, therapeutic index means small changes in dose, drug interactions, kidney function or liver metabolism can cause toxicity or treatment failure, so these drugs need therapeutic drug monitoring and careful, small dose adjustments.

How is therapeutic index different from potency and efficacy?

Potency is how much drug is needed for an effect, shown by where the dose-response curve sits on the dose axis; a lower ED50 means a more potent drug. Efficacy is the maximal response the drug can produce, the height of the curve's plateau. The therapeutic index compares toxic and effective doses. A drug can be very potent yet have a narrow therapeutic index, so potency never tells you how safe a drug is.

Why do narrow therapeutic index drugs need monitoring?

Because effective and toxic levels are close together, anything that changes drug exposure can push a patient out of range. Classic examples are lithium toxicity after starting a thiazide, ACE inhibitor or NSAID, theophylline toxicity with macrolide antibiotics, digoxin toxicity with low potassium or magnesium, and large phenytoin level jumps after small dose increases. Monitoring uses drug levels or effect markers, such as the INR for warfarin.

Therapeutic Index: Formula, Narrow TI Drugs & Exam Traps

Q: Which drugs have a narrow therapeutic index?

Classic narrow therapeutic index drugs include warfarin, digoxin, lithium, phenytoin, carbamazepine, theophylline, levothyroxine and the transplant drugs tacrolimus, cyclosporine and sirolimus. FDA materials give examples rather than a complete list, naming warfarin, levothyroxine, tacrolimus, sirolimus, phenytoin and carbamazepine, and Health Canada's critical-dose list adds digoxin, lithium, theophylline, cyclosporine and flecainide. A study mnemonic is Warning! These Drugs Are Lethal! (warfarin, theophylline, digoxin, antiepileptics, lithium). Aminoglycosides and vancomycin are also dosed by blood levels in hospital.

The therapeutic index (TI) is a drug's safety margin: TI = TD50/ED50, the median toxic dose divided by the median effective dose (LD50/ED50 in animal studies). A higher TI means a safer drug. A narrow TI means the toxic and effective doses sit close together, so a small change in dose, metabolism or kidney function can tip a patient into toxicity or treatment failure. Classic narrow therapeutic index (NTI) drugs include warfarin, digoxin, lithium, phenytoin, carbamazepine, theophylline, levothyroxine and the transplant drugs tacrolimus, cyclosporine and sirolimus, and they are dosed with therapeutic monitoring (drug levels, or markers such as the INR). TI is not potency (how much drug you need) and not efficacy (the biggest effect the drug can produce).

The definitions you need

$\text{TI} = \frac{TD_{50}}{ED_{50}}$

Term	Meaning	Exam point
ED50	Dose that produces the desired effect in 50% of a population	Lower ED50 = more potent
TD50	Dose that produces a defined toxic effect in 50% of a population	Read off the toxicity curve
LD50	Dose lethal to 50% of test animals	Animal toxicology only; replaces TD50 in animal studies
Therapeutic index	TD50/ED50 (or LD50/ED50)	A ratio without units; higher = safer
Therapeutic window	Concentration range between the minimum effective and the minimum toxic concentration	The clinical version, used with drug levels
NTI drug	Small differences in dose or blood concentration may cause serious therapeutic failure or life-threatening adverse reactions (FDA)	Needs monitoring and careful titration

Three wordings you may meet:

Pharmacology textbooks and review books: TD50/ED50 from population (quantal) data.
Merck Manual: "the ratio of the minimum toxic concentration to the median effective concentration." Same idea: the bigger the ratio, the more room between help and harm, and raising the dose of a drug with a small TI raises the chance of toxicity.
US regulation (21 CFR 320.33(c)): among its criteria for possible bioequivalence problems, it gives as an example of a "narrow therapeutic ratio" less than a 2-fold difference between LD50 and ED50, or between the minimum toxic and minimum effective blood concentrations, when safe use requires careful dose titration and patient monitoring.

The FDA describes NTI drugs by five features: little separation between therapeutic and toxic doses; concentrations that are too low or too high can cause serious therapeutic failure or side effects; therapeutic monitoring; low within-subject variability; and dose changes in very small steps. Generic NTI drugs must meet tighter quality and bioequivalence standards than other generics. FDA materials give examples rather than a complete list: warfarin and levothyroxine (CDER), plus tacrolimus, sirolimus, phenytoin and carbamazepine in an FDA presentation; Health Canada's "critical dose" list also includes digoxin, flecainide, lithium, theophylline and cyclosporine.

Reading TI off dose-response curves

A quantal curve plots dose (usually on a log scale) against the percentage of a population showing an all-or-none response, such as "seizures controlled" or "arrhythmia occurs." Draw two S-shaped curves: the efficacy curve on the left and the toxicity curve to its right.

ED50 is where the efficacy curve crosses 50%; TD50 is where the toxicity curve crosses 50%.
Wide gap between the curves = high TI. Shifting the toxicity curve right makes a drug safer; shifting it left narrows the TI.
A graded curve plots the size of the response in one patient or tissue. It shows potency (where the curve sits on the dose axis), maximal efficacy (the plateau) and slope (change in response per unit dose), the three features Merck lists, but one graded curve can't give you TI.
TI ignores the tails. It compares the 50% points only. If the curves are shallow and overlap, some patients get toxicity at doses that are barely effective for others.

Here is a hypothetical population study:

Dose (mg)	% with the desired effect	% with toxicity
1	10	0
2	50	0
5	90	2
20	99	20
40	100	50
80	100	85

ED50 = 2 mg and TD50 = 40 mg, so TI = 40/2 = 20. Notice the tail problem: the 20-mg dose helps 99% of people but is toxic in 20%. More on curve shapes in the dose-response curve tips.

Worked calculations

These numbers are illustrative, not data for real drugs.

1. Two drugs, potency vs safety. Drug A: ED50 = 10 mg, TD50 = 1,000 mg, so TI = 100. Drug B: ED50 = 0.5 mg, TD50 = 1.5 mg, so TI = 3. Drug B is 20 times more potent (0.5 mg vs 10 mg) but far less safe. Potency tells you the dose, not the margin.

2. Animal data. LD50 = 300 mg/kg and ED50 = 3 mg/kg, so TI = 100. You will never see an LD50 measured in people.

3. A real therapeutic window. For phenytoin, the label says therapeutic effect without toxicity occurs more often at total serum concentrations of 10-20 mcg/mL. Nystagmus on lateral gaze usually appears at about 20 mcg/mL and ataxia at about 30 mcg/mL. Because its metabolism saturates, small dose increases near the top of the range can produce very large jumps in level (see zero-order elimination).

Potency vs efficacy vs therapeutic index

Concept	Question it answers	Where to look	Trap
Potency	How much drug for a given effect?	Position along the dose axis (ED50 or EC50); left = more potent	More potent does not mean safer or better
Efficacy	How big can the effect get?	Height of the plateau (maximal response)	A partial agonist has lower efficacy than a full agonist, whatever its potency
Therapeutic index	How far apart are help and harm?	Distance between the efficacy and toxicity curves	Needs both an effective and a toxic dose

Agonist types and their curves are in the agonist vs antagonist mnemonic.

Why narrow-TI drugs cause trouble

The wanted effect is the danger. More anticoagulation means more bleeding with warfarin.
Clearance changes. Digoxin and aminoglycosides become riskier as kidney function falls (Merck); lithium levels climb with sodium loss and with thiazides, ACE inhibitors or NSAIDs.
Metabolism changes. Theophylline levels change with macrolides and other drugs metabolized by CYP450, and with fever, liver disease or heart failure (Merck). Carbamazepine is itself a potent CYP450 inducer.
Saturable kinetics. Phenytoin's metabolism saturates, so levels rise out of proportion to the dose.
Protein binding. With low albumin or renal or hepatic disease, the phenytoin label says unbound levels may be more relevant than total levels.
Electrolytes. With hypokalemia or hypomagnesemia, digoxin can be toxic at lower doses and serum levels (Merck).

Narrow-TI drugs and how they are monitored

Drug	Monitor	Toxicity clues	Watch for
Warfarin	INR	Bleeding	Interacting drugs (CYP450 inducers and inhibitors)
Digoxin	Serum level, potassium, magnesium, renal function	Nausea, vomiting, anorexia, confusion, blurred vision (amblyopia); VT, VF, complete heart block	Renal dysfunction, hypokalemia, hypomagnesemia, older age, low lean body mass, amiodarone
Lithium	Serum level, BUN and creatinine, TSH, calcium and PTH	Gross tremor, brisk reflexes, headache, vomiting, confusion, then stupor, seizures, arrhythmias	Thiazides, ACE inhibitors, NSAIDs other than aspirin; sodium loss from fever, vomiting, diarrhea or diuretics
Phenytoin	Total level; unbound level if albumin is low	Nystagmus, ataxia, dysarthria, lethargy, confusion; chronic gingival hyperplasia, hirsutism	Small dose increases near the top of the range
Carbamazepine	Levels	Diplopia, dizziness, nystagmus, GI upset	Potent CYP450 inducer: lowers levels of other drugs
Theophylline	Serum level, kept at 5-15 mcg/mL (Merck)	Headache, vomiting, arrhythmias, seizures	Macrolides and other CYP450 drugs; fever, liver disease, heart failure
Aminoglycosides	Peak and trough after the 2nd or 3rd dose; creatinine every 2-3 days	Nephrotoxicity (often reversible); vestibular and hearing damage (often irreversible); prolonged neuromuscular blockade	High levels, courses over 3 days, older age, kidney disease, other nephrotoxins; loop diuretics for hearing loss
Vancomycin	AUC-guided dosing: target AUC 400-600 mg·h/L for serious MRSA infection (assuming MIC 1 mg/L)	Nephrotoxicity	Trough-only targets of 15-20 mg/L were linked to more kidney injury (2020 consensus guideline)

Levothyroxine is titrated to the TSH, and tacrolimus, cyclosporine and sirolimus are dosed by trough blood levels. Aminoglycosides and vancomycin are not on the regulatory NTI lists above, but they are classic level-guided hospital drugs. A study mnemonic (an unofficial memory aid, not an FDA list): "Warning! These Drugs Are Lethal!" = Warfarin, Theophylline, Digoxin, Antiepileptics (phenytoin, carbamazepine), Lithium. Lithium monitoring in depth is in the lithium guide.

How USMLE tests it: the traps

Upside-down ratio. TI = TD50/ED50: toxic on top. ED50/TD50 is a distractor.
"Low TI" = "narrow TI" = unsafe. A higher number is the safer drug.
Potency is not safety. The most potent drug in a table is often the one with the narrowest TI.
Ratio vs range. TI is a ratio of doses; the therapeutic window is a range of concentrations.
LD50 is an animal number.
The stable patient who becomes toxic. A patient on lithium starts a thiazide or NSAID; a patient on theophylline starts a macrolide; a patient on digoxin becomes hypokalemic on a diuretic. The question is really asking about a narrow TI.
Phenytoin: a small dose increase causes a disproportionate rise; with low albumin the total level can mislead.
Digoxin toxicity at a "normal" level when potassium or magnesium is low.

Practice question

A population study compares four investigational drugs for the same indication. The median effective dose (ED50) and median toxic dose (TD50) of each are shown:

Drug	ED50 (mg)	TD50 (mg)
W	2	8
X	20	400
Y	0.5	1
Z	50	250

Which drug has the highest therapeutic index?

A. Drug W B. Drug X C. Drug Y D. Drug Z E. It cannot be determined without each drug's maximal efficacy

Answer: B. TI = TD50/ED50 = 400/20 = 20, the largest gap between the toxic and effective doses.

A: 8/2 = 4.
C: 1/0.5 = 2. Drug Y is the most potent (lowest ED50) but has the narrowest TI, the classic trap. It is the one most likely to need therapeutic drug monitoring.
D: 250/50 = 5. It is the least potent, but potency does not decide safety.
E: TI needs only TD50 and ED50. Maximal efficacy is a separate property, read from the plateau of a graded dose-response curve.

Key takeaways

TI = TD50/ED50 (LD50/ED50 in animals); a higher TI is safer.
The therapeutic window is the concentration range between the minimum effective and minimum toxic levels.
Classic NTI drugs: warfarin, digoxin, lithium, phenytoin, carbamazepine, theophylline, levothyroxine, tacrolimus, cyclosporine, sirolimus.
NTI drugs need therapeutic monitoring and small dose steps; levels shift with kidney function, CYP450 interactions, saturable kinetics, albumin and electrolytes.
Potency = position on the dose axis; efficacy = plateau height; TI = distance between the efficacy and toxicity curves.
In vignettes, a new interacting drug or falling kidney function in a stable patient on an NTI drug points to toxicity.