Pharmacokinetics,
derived rather than memorised.

Pharmacodynamics asks what the drug does to the body. Pharmacokinetics asks the reverse question: what the body does to the drug. Formally, it is the study of the distribution of drugs in the various compartments of the body and the changes that occur with time. Every section of this module is ultimately an answer to a single question — given a dose administered at time zero, what is the drug concentration at the site of action at time t?

The classical description of the journey a drug makes is LADME: liberation of drug from its formulation, absorption into the systemic circulation, distribution to the tissues, metabolism into more polar compounds, and excretion from the body. Metabolism and excretion are usually considered together as elimination, since both remove drug irreversibly.

More useful for examination purposes, however, is a different classification — one that separates the parameters describing genuine physiological properties from those that are merely arithmetic consequences of them. This distinction determines how nearly every dosing question should be answered.

Clearance and volume of distribution are independent of one another and of everything else. They are properties a patient possesses. Half-life is not: it is a number computed from the other two, and it can change because either of them has changed. A prolonged half-life therefore has two possible causes with opposite management, a point developed fully in section 03.

Take blood samples after an intravenous bolus and plot concentration against time. The concentration falls, but not in a straight line: it falls quickly at first and progressively more slowly, approaching but never quite reaching zero. This shape is not incidental. It arises directly from the physical nature of elimination, and understanding why is the foundation of everything that follows.

Why decay is exponential

Consider what the liver and kidney actually do. A fixed volume of blood is delivered to the organ each minute, and the organ removes drug from that blood. It does not remove a fixed quantity of drug, because it cannot remove drug that is not presented to it. It removes whatever is contained in the blood that arrives. Therefore, when the plasma concentration is high, more drug is removed per minute; when it is low, less is removed.

Expressed formally, the rate at which the concentration falls is proportional to the concentration itself. That single statement is the definition of an exponential process. Writing the rate of change as the derivative dC/dt:

The minus sign indicates a falling quantity. The constant of proportionality k is called the elimination rate constant. Because the derivative depends on C raised to the power one, this is described as a first-order relationship, and hence first-order kinetics. The terminology comes from the exponent, not from any notion of the reaction being “first” in a sequence.

Solving the equation

The differential equation tells us the rate at every instant, but what we need is a formula for the concentration itself at any time. Obtaining it requires integration. The technique is separation of variables: rearrange so that all terms in C are on one side and all terms in t on the other, then integrate both sides.

The elimination rate constant, k

A rate constant of 0.2 h⁻¹ means that 20% of whatever drug is present is removed each hour — not 20% of the original dose, but 20% of what remains at that moment. This is why the absolute quantity removed falls continuously while k itself does not change. Note also that k is a rate constant, not a rate: the rate of elimination is k × C, and it changes constantly.

The time constant, τ

The first definition is the one to give in a viva, because it explains the graphical construction. Draw a tangent to the curve at time zero, where the process is steepest, and continue that tangent until it crosses the horizontal axis. It does so at t = τ. The process is of course not complete at that point — it is only 63.2% complete — precisely because the rate did not remain at its initial value but slowed continuously. The gap between the tangent and the real curve is a visual measure of how exponential the process is.

The half-life, t½

Where 0.693 comes from

The number 0.693 appears throughout pharmacokinetics and is usually memorised without explanation. It is simply the natural logarithm of 2, and it arises the moment the word “half” enters the algebra. Substituting the defining condition of half-life — that C has fallen to C₀/2 — into the exponential equation:

Relating the three

Since τ = 1/k, substituting into t½ = 0.693/k gives:

The time constant is always longer than the half-life, by a factor of 1/ln 2 ≈ 1.44. The reason is intuitive once stated: e (2.718) is greater than 2, so falling to 1/e of a value takes longer than falling to 1/2 of it.

Completion of an exponential process

A first-order process is, strictly, never complete — it approaches zero asymptotically. For practical purposes it is regarded as complete after 4–5 half-lives, or after 3 time constants.

The three exponential functions

Not every exponential falls. Three forms occur in anaesthetic practice, and they should be distinguished by their differential equations rather than memorised as shapes.

Note that the wash-in curve is still governed by a negative exponential, despite rising. What decays exponentially is not the quantity but the distance remaining to the plateau. This is why wash-in and wash-out share the same time constant and the same “95% complete at 3τ” rule.

Absorption is preceded by liberation, the release of drug from its pharmaceutical formulation. This is rate-limiting for slow-release and enteric-coated preparations, and irrelevant for a solution given intravenously. Factors affecting liberation are the formulation itself and the route chosen.

Two properties of absorption must be distinguished throughout, because they are separately measurable and separately important:

• The rate of absorption, described by the peak concentration reached (C_max) and the time taken to reach it (T_max). Rate determines speed of onset.
• The extent of absorption, the proportion of the dose that eventually gets in. Extent determines the size of the effect and the steady-state concentration.

A slow-release preparation and an immediate-release preparation of the same drug may have identical extent but very different rates. Conversely, vomiting reduces extent without necessarily altering the rate at which what remains is absorbed.

Determinants of the extent of absorption

For an orally administered drug, loss can occur at three places, and a structured answer should address each in turn.

Drug factors that independently impair gastrointestinal absorption are high molecular weight and poor lipid solubility — glycopyrrolate, neostigmine, aminoglycosides and the neuromuscular blocking drugs are all poorly absorbed for this reason — together with slow dissolution and instability in gastrointestinal secretions. Patient factors are gastric emptying rate, gastrointestinal transit time, gut pH and splanchnic blood flow.

Routes of administration

The rectal route

The rectal route deserves separate treatment because its anatomy explains its unpredictability. The proximal rectum is drained by the superior haemorrhoidal vein into the inferior mesenteric vein and thence into the portal vein — so drug absorbed here undergoes first-pass metabolism. The distal rectum is drained by the inferior haemorrhoidal vein, which is a systemic vein and bypasses the liver. Since a suppository cannot be reliably placed at a chosen depth, and may migrate, roughly half the absorbed dose bypasses the liver but the exact proportion is unknowable.

Absorption occurs by passive diffusion, governed by Fick’s law, and is limited by the small surface area of the rectal mucosa. It is further reduced by the presence of faecal material, by metabolism by intestinal flora, and by diarrhoea. Bioavailability is therefore variable and generally lower than oral, so doses two to ten times larger may be required.

Its advantages are that it can be used when no intravenous access is available, in unconscious patients and in fasting patients, and that it causes less nausea and vomiting. Drugs given rectally include barbiturates, diazepam, ketamine, midazolam, paracetamol and diclofenac.

First-pass pulmonary uptake

Frequently forgotten. The lung is the first capillary bed an intravenously administered drug encounters, and it takes up lipophilic amines with a pK_a above about 8 substantially, blunting the peak arterial concentration and acting as a reservoir from which drug is subsequently released. Uptake exceeding 65% of the dose occurs with lignocaine (65%), pethidine (65%), propranolol (75%) and fentanyl (75%), and also with alfentanil and sufentanil. The magnitude of the effect is not altered by spontaneous respiration, controlled ventilation or apnoea.

The cell membrane is approximately 10 nm thick, a bimolecular leaflet of amphipathic lipids with hydrocarbon chains oriented inward and polar head groups outward, studded with proteins.

Ionisation

Only the unionised form of a drug is sufficiently lipid soluble to diffuse across a membrane. Ionised molecules carry charge, are surrounded by a shell of water, and are effectively excluded from the lipid phase. The fraction unionised at a given pH therefore governs how readily a drug crosses the gut wall, the blood–brain barrier, the placenta and the renal tubular epithelium.

The relationship between pH, pK_a and the ratio of ionised to unionised drug is the Henderson–Hasselbalch equation. Its two forms are mirror images, and the commonest error is to apply the acid form to a base.

Rearranging gives the forms most useful in an examination, and note that the exponent is identical in both — it is the labels that swap:

Writing Δ = pH − pK_a, the percentage unionised follows directly:

Ion trapping

The mechanism reduces to one sentence: only the unionised species crosses; once across, it re-equilibrates to the local pH; if the local pH favours ionisation, the molecule is converted into a form that cannot cross back. The consequence is that the gradient of total drug is maintained indefinitely, while the gradient of unionised drug is zero. Equilibrium exists, but only for the diffusible species.

• Stomach. Basic drugs concentrate in the strongly acidic gastric lumen.
• Placenta. Fetal pH is lower than maternal. Unionised local anaesthetic crosses, becomes ionised in the more acidic fetal compartment and is trapped, maintaining a gradient for continued transfer. Fetal acidosis worsens accumulation, so the effect is greatest in the distressed fetus that can least tolerate it.
• Renal tubule. Alkalinising the urine ionises weak acids and prevents their passive reabsorption — the basis of urinary alkalinisation in salicylate poisoning.
• Breast milk is more acidic than plasma, so basic drugs may be concentrated in it.

The phrase “as intact drug” is doing substantial work. A drug that is completely absorbed from the gut but wholly metabolised on its first passage through the liver has an absorption of 100% and a bioavailability of zero. Glyceryl trinitrate given orally is the standard example of near-zero oral bioavailability, which is why it is given sublingually.

Why bioavailability is measured using area under the curve

This is the part most often recited without understanding. The reasoning is worth following, because it also explains why C_max cannot be used instead.

The problem is to determine how much drug got in. Plasma concentration cannot answer this directly, because concentration at any single moment reflects both how much has arrived and how much has already been removed. What is needed is a measure of total systemic exposure over all time, and that is precisely what the integral of the concentration–time curve provides. The area under the curve, AUC, has units of concentration × time and represents cumulative exposure.

This expression contains clearance, which is not usually known. The elegant step is to compare two routes in the same person, in whom clearance is the same. Writing the relationship once for the intravenous dose (where F = 1) and once for the oral dose, then dividing one by the other, clearance cancels:

Where equal doses are compared, the second bracket becomes one and the expression reduces to the familiar form:

Measuring AUC in practice

AUC is estimated using the trapezoid rule: the area is divided into trapezoids between successive sampling times and summed. Its advantage is that it makes no assumption about the underlying pharmacokinetic model. Its disadvantages are that it assumes a straight line between data points, so where the curve is steep the error may be appreciable, and whether the estimate is too high or too low depends on whether the curve is rising or falling at that point.

Three related terms

Bioavailability and the extraction ratio

For a drug that is completely absorbed from the gut, the fraction escaping hepatic first-pass metabolism is (1 − ER), where ER is the hepatic extraction ratio. Combining all three sites of loss:

It follows immediately that a drug with a high hepatic extraction ratio can never have a high oral bioavailability, however completely it is absorbed. This is the quantitative link to the extraction ratio developed in section 12, and the connection examiners have specifically asked candidates to make.

The word apparent carries the entire concept. V_d is not an anatomical space that could be dissected or measured with a dye. It is a proportionality constant relating the dose administered to the concentration observed. If a drug is sequestered in fat, very little remains in plasma, the measured concentration is low, and the arithmetic returns a volume larger than the patient. Thiopentone has a V_d of approximately 3 L·kg⁻¹, which in a 70 kg adult is 210 litres — physically impossible, and pharmacologically informative, because it proves extensive extravascular sequestration.

Landmark volumes

Interpretation requires knowing the real compartments a 70 kg adult possesses: plasma ≈ 3 L, blood 5 L, extracellular fluid 14 L, total body water 42 L. A drug with a V_d of 5–20 L is effectively confined to plasma and extracellular fluid. A V_d near 42 L indicates even distribution through total body water. Anything substantially above 42 L proves tissue binding, because no anatomical compartment of that size exists.

Determinants

1. Lipid solubility. Lipophilic drugs partition into fat and have large volumes.
2. Degree of ionisation, itself a function of pK_a and ambient pH. Ionised drug remains in plasma and extracellular fluid.
3. Molecular size. Large molecules cannot cross the vascular endothelium — dextran is the classic illustration.
4. Plasma protein binding. Binding retains drug in plasma and reduces V_d. Propofol is the instructive exception: 98% protein bound yet V_d 3.5–4.5 L·kg⁻¹, because its lipid solubility overwhelms its binding.
5. Tissue binding. Drug bound to muscle or adipose tissue increases V_d — amiodarone in fat, iodine in thyroid, tetracycline in bone.
6. Regional blood flow, altered by age, cardiac failure, chronic renal failure and hypovolaemic shock.
7. Timing of sampling, which is a methodological rather than physiological determinant — see Figure 4.

Measuring V₁, and why the answer depends on when you sample

The procedure is: give an intravenous bolus, sample plasma at intervals, plot log concentration against time, extrapolate the terminal straight portion back to time zero to obtain C₀, and divide the dose by it. C₀ is the concentration that would have been observed had distribution been instantaneous.

The peripheral volume is obtained from the intercompartmental rate constants: V₂ = V₁(k₁₂ + k₂₁)/k₂₁ = V₁(1 + k₁₂/k₂₁).

Representative values

Clinical significance

1. It determines the loading dose: LD = V_d × C_target. Note that clearance does not appear.
2. It is one of the two determinants of half-life.
3. It predicts the suitability of a drug for extracorporeal removal. A drug with a large V_d is mostly outside the plasma and therefore cannot be removed effectively by dialysis or haemofiltration, however efficient the filter.

The unbound fraction is determined by the affinity of the drug for the protein, the concentration of the binding protein, and the concentration of drug relative to that of the protein.

An important qualification: although basic drugs bind preferentially to α₁-acid glycoprotein, albumin carries a greater absolute fraction of most drugs simply because its plasma concentration is far higher.

Why displacement interactions are usually unimportant

The intuitive expectation is that displacing a drug from protein raises the free concentration and produces toxicity. In most cases it does not, and the reason is worth following.

Displacement does raise free concentration acutely. But the newly freed drug is immediately exposed to both distribution into tissues and clearance by the liver and kidney. A new steady state is reached in which the free concentration has returned to its previous value while the total concentration is lower. The active concentration is therefore unchanged.

Displacement becomes clinically significant only when three conditions coincide: the drug is highly protein bound, has a small volume of distribution, and has a narrow therapeutic index. Warfarin satisfies all three — 99% bound, V_d about 10 L. If binding falls from 99% to 98%, the free fraction doubles, a 100% increase in active drug. Compare morphine at 20–40% bound, where the same absolute change is trivial. Aspirin raises the free fraction of warfarin from about 1% to 5%.

Protein binding across the placenta — a worked illustration

Consider a drug present in maternal blood at a total concentration of 100 ng·mL⁻¹ with 90% protein binding, so the maternal free concentration is 10 ng·mL⁻¹. Only free drug crosses the placenta, so at equilibrium the fetal free concentration is also 10 ng·mL⁻¹. If fetal protein binding is only 10%, the fetal total concentration is approximately 11 ng·mL⁻¹.

The maternal:fetal total ratio is therefore about 9:1, while the ratio that matters pharmacologically — free drug — is 1:1. Equilibrium is always of free drug, never of total drug. Quoted maternal:fetal ratios are meaningless unless the binding on each side is also stated.

Effects on clearance

Protein binding affects clearance differently at each organ, and the pattern follows directly from the extraction ratio logic developed in section 12:

• High hepatic extraction ratio: clearance is insensitive to protein binding, because the hepatocyte strips drug from protein during a single transit.
• Low hepatic extraction ratio: clearance is proportional to the free fraction, because only free drug reaches the enzyme and extraction is slow enough for binding to be rate-limiting.
• Glomerular filtration: dependent on protein binding — only free drug is filtered.
• Active tubular secretion: independent of protein binding — the transporter is avid enough to strip bound drug, exactly as a high-extraction liver does.

The purpose clause explains why metabolism is necessary at all. A lipophilic drug filtered at the glomerulus would simply be reabsorbed passively across the tubular epithelium and returned to the circulation — it could never be excreted. Metabolism converts it into a form the kidney can retain in the tubular lumen. Metabolism generally, though not invariably, reduces biological activity, and increased polarity also limits the metabolite’s access to receptor sites.

Sites of metabolism

• Liver — the great majority.
• Plasma and tissue esterases — suxamethonium and mivacurium (plasma cholinesterase), esmolol (red-cell esterase), remifentanil (non-specific tissue esterases, particularly in muscle).
• Gut wall — chlorpromazine, isoprenaline. Kidney — midazolam, dopamine. Lung — angiotensin I, prilocaine.
• Hofmann elimination — atracurium; spontaneous and non-enzymatic, dependent on temperature and pH, and therefore independent of hepatic and renal function.

Phase I and Phase II

Cytochrome P450

A superfamily of haemoproteins forming the terminal oxidases of the mixed-function oxidase system, located in the smooth endoplasmic reticulum. The name derives from the observation that, in the reduced state, they combine with carbon monoxide to form a complex with maximal light absorption at 450 nm. The CYP1, CYP2 and CYP3 families account for about 70% of total hepatic P450 and for most Phase I reactions in humans. During oxidation the oxidised enzyme combines with drug to form a complex, which is then reduced by cytochrome P450 reductase with transfer of an electron from NADPH.

Induction and inhibition

Before memorising lists, understand the mechanistic difference, which is more examinable than either list.

Drugs whose metabolism is notably affected by these interactions include warfarin, phenytoin, theophylline, ciclosporin, bupivacaine, lignocaine, diazepam, nifedipine, sildenafil, vecuronium, imipramine and quinidine — a list dominated, as predicted by section 12, by low extraction ratio drugs.

Pharmacogenetics

• Acetylator status is genetically determined; approximately 50% of Caucasians and 90% of Japanese people are fast acetylators. Relevant to isoniazid, hydralazine, procainamide, sulphonamides and phenelzine. Procainamide induces antinuclear antibodies and a lupus-like syndrome more rapidly in slow acetylators.
• CYP2D6 — codeine and tramadol are prodrugs requiring conversion to morphine and O-desmethyltramadol respectively. Poor metabolisers obtain little analgesia; ultra-rapid metabolisers are at risk of opioid toxicity.
• CYP2C19 — clopidogrel is a prodrug activated by this enzyme. Carriers of a variant allele have a substantially higher risk of thromboembolic complications; prasugrel is an alternative.
• Plasma cholinesterase variants prolong the action of suxamethonium and mivacurium.

Prodrugs and toxic metabolites

Glomerular filtration rate — definitions

The word ultrafiltrate matters. The glomerular filtration barrier — fenestrated endothelium, basement membrane and podocyte slit diaphragms — is both size- and charge-selective, so the filtrate is essentially protein-free. This is the anatomical basis of the pharmacokinetic rule that only unbound drug is filtered: drug bound to albumin cannot cross a barrier that retains albumin itself.

The formula for GFR

GFR is determined by the balance of Starling forces across the glomerular capillary, multiplied by the permeability and surface area of the barrier:

Substituting the typical values gives a net filtration pressure of (45 − 10) − (25 − 0) = 10 mmHg, and therefore GFR = 12.5 × 10 = 125 mL·min⁻¹.

In practice GFR is not calculated from Starling forces but measured as the clearance of a marker substance that is freely filtered and neither secreted nor reabsorbed — inulin is the reference standard:

Note that this is exactly the clearance equation of section 11 applied to a substance chosen so that its clearance equals the filtration rate — which is why GFR has the units of a clearance. The filtration fraction, GFR ÷ renal plasma flow, is normally about 0.2 (125 ÷ 650 mL·min⁻¹).

Multiply by 0.85 for females. To convert a reported eGFR to an absolute value for dosing purposes, multiply by the patient’s actual body surface area and divide by 1.73.

The four renal processes

1 · Glomerular filtration

• Only the free, unbound drug in plasma water is filtered — filtration is therefore dependent on protein binding.
• Determined by molecular size (below approximately 50 000 Da), electrical charge, the number of functioning nephrons and the GFR itself.
• Filtration handles polar, water-soluble drugs and metabolites; it is the reason metabolism must first render lipophilic drugs polar.
• If renal clearance equals GFR, the drug has a low renal extraction ratio.

2 · Active tubular secretion

• Occurs in the proximal tubule by active carrier-mediated transport, requires cellular energy, and can proceed against a considerable concentration gradient.
• So avid that protein-bound drug is stripped from albumin during a single pass — the renal equivalent of a high hepatic extraction ratio. Secretion is therefore independent of protein binding but dependent on renal perfusion.
• Two separate, distinct and saturable transport systems exist: one for organic acids (which also secretes uric acid; handles penicillins, cephalosporins, salicylates, thiazides, frusemide, glucuronides) and one for organic bases (dopamine, morphine, neostigmine, lignocaine).
• Drugs within a system compete. Probenecid and penicillin is the classical pair, deliberately exploited to prolong penicillin levels.
• Renal clearance approaches effective renal plasma flow — a high extraction ratio.

3 · Passive tubular reabsorption

• Occurs in the distal tubule by passive non-ionic diffusion of the lipid-soluble unionised species back into plasma.
• Urine pH ranges from 4.5 to 8.0, so a considerable hydrogen ion gradient can exist between urine and plasma.
• Acidic drugs are excreted faster in alkaline urine; basic drugs are excreted faster in acidic urine.
• Depends on urine pH, which determines the unionised fraction, and on tubular flow rate, which determines transit time.

4 · Tubular diffusion and ion trapping

This is the component most often left out of the four. Drug secreted proximally enters the tubular lumen as a non-diffusible ion. As the urine acidifies distally, an acidic drug becomes unionised and diffuses back into plasma, reducing its excretion. A basic drug remains ionised, is trapped as a cation, and the resulting fall in luminal unionised concentration establishes a further gradient drawing more unionised drug out of the plasma — actively enhancing its excretion.

Renal impairment and dose adjustment

• When renal clearance exceeds 70% of total clearance, accumulation occurs in renal failure — aminoglycosides, digoxin, chlorpropamide, pancuronium, d-tubocurarine, gallamine, alcuronium.
• When renal clearance is below 30% of total clearance, renal disease has little effect.
• Adjustment is generally required only when a drug is more than 50% renally cleared and renal function is reduced to half of normal or less.
• Remember active metabolites: morphine-6-glucuronide, norpethidine and desmethyldiazepam all accumulate in renal failure even though the parent drug is hepatically metabolised.
• Drugs with organ-independent elimination are preferred: atracurium (Hofmann elimination) and remifentanil (non-specific esterases).

Biliary excretion and enterohepatic circulation

• Usually the major route for compounds with molecular weight above 400–500 Da. Because Phase II conjugation substantially increases molecular weight, metabolism can switch a drug from renal to biliary elimination.
• Transport is active, saturable and non-specific, with separate systems for acids and bases.
• Can concentrate drug up to 100 times plasma levels — the basis of biliary contrast imaging and of high ampicillin concentrations in enteric fever.
• High biliary clearance: acidic — ampicillin, rifampicin, radiographic contrast; basic — vecuronium, pancuronium, alcuronium, glycopyrrolate.
• Enterohepatic circulation: glucuronide conjugates secreted in bile are hydrolysed by bacterial glucuronidase in the small intestine; the freed drug is reabsorbed, re-metabolised and re-excreted, prolonging the drug’s presence. Broad-spectrum antibiotics destroy the responsible flora, reduce recirculation and enhance elimination — the mechanism of oral contraceptive failure.

Other routes

Quantitatively unimportant but examinable, and specifically requested in the April 2024 paper: lungs (volatile agents), bile and faeces (vecuronium), sweat, saliva, tears and breast milk. All depend on diffusion of the unionised lipid-soluble form and are therefore pH-dependent. Breast milk is more acidic than plasma, so basic drugs may be concentrated there. Hair and skin permit forensic detection of toxic metals such as arsenic and mercury.

These are the same statement, and demonstrating that they are is worth doing in a viva because it shows understanding rather than recall. Take definition (ii) and substitute dimensions:

Because clearance is a flow, it has the same dimensions as cardiac output or glomerular filtration rate. This is not coincidental: for a substance completely removed by the kidney in a single pass, renal clearance equals renal plasma flow.

Clearance is additive

Two physiological variables control the clearance achieved by any organ: the efficiency of that organ at extracting drug, and the blood flow delivered to it. Which of the two dominates is the entire content of section 12.

Measuring clearance

1 · From a single dose — Cl = Dose ÷ AUC

This relationship was derived in section 06 while establishing bioavailability, where it emerged that F × Dose = Cl × AUC. For an intravenous dose, F = 1, so:

It is worth restating why this holds so generally. The derivation invoked only two assumptions: that the rate of elimination is Cl × C, and that everything which entered eventually leaves. It made no assumption about the number of compartments. Consequently Cl = Dose ÷ AUC applies to one-compartment models, multicompartment models and any pattern of intravenous dosing, provided the total dose is used and clearance is constant. It follows immediately that AUC is proportional to dose in linear kinetics — a property that fails under saturation and can therefore be used to detect it.

2 · Renal clearance

Directly measurable, which is why renal clearance is the only clearance genuinely known rather than inferred.

3 · From an infusion at steady state

Clinical applications

Clearance determines the maintenance dose. At steady state, by definition, the rate of administration equals the rate of elimination:

Clearance is also one of the two determinants of half-life, through t½ = 0.693 V_d ÷ Cl. The derivation of this relationship requires the bridging identity Cl = k × V_d, obtained as follows:

An extraction ratio of 1 means every molecule presented is removed in one pass; a ratio of 0 means none is. Since clearance is the volume completely cleared per unit time, and blood flow Q is the volume presented per unit time, the two are linked by the simplest possible relationship:

The three determinants of hepatic clearance

1. Hepatic blood flow (Q) — approximately 1500 mL·min⁻¹, around 25% of cardiac output, of which 75% arrives via the portal vein and 25% via the hepatic artery.
2. Intrinsic clearance (Cl_i) — the maximum ability of the liver to remove drug irreversibly by metabolism or biliary excretion. It reflects enzyme activity, is independent of blood flow, and has a unique value for each drug.
3. The unbound fraction (f_u) — the proportion of drug in blood not bound to plasma protein.

The relationship between them is the well-stirred model:

This single equation contains the whole of high- and low-extraction pharmacology. Taking its two limits:

Why a high extraction ratio drug is sensitive only to blood flow

The algebra above states the result; the following explains it. Consider a drug whose hepatocytes are extremely avid — the enzymes have vastly more capacity than the drug delivery ever requires. Such a liver removes, say, 95% of everything presented to it.

Why does doubling enzyme activity achieve nothing? Because the liver is already removing almost everything it receives. It cannot remove more than 100% of what arrives. Doubling the enzyme capacity might raise extraction from 95% to 97.5%, a change of no consequence. The enzymes are not the bottleneck — delivery is.

Why does protein binding not matter? Because the drug–protein complex is in rapid reversible equilibrium with free drug. As the hepatocyte removes free drug, more dissociates from albumin immediately to restore the equilibrium. Extraction is so avid that, within the transit time of a single pass, drug is effectively stripped from its binding protein. The hepatocyte therefore sees the total concentration, not the free concentration.

What then limits clearance? Only the amount of drug arriving per minute, which is Q × C_a. Since clearance is defined as the volume cleared per minute, and essentially every millilitre arriving is cleared, clearance is simply the blood flow. Halve hepatic blood flow and you halve clearance.

Why a low extraction ratio drug is sensitive to enzyme activity and protein binding

Now consider the opposite drug, whose enzymes are slow relative to delivery. Such a liver removes perhaps 10% of what is presented.

Why does increasing blood flow achieve nothing? Because drug is already arriving far faster than the enzymes can process it. Doubling the flow doubles the delivery, but the enzymes still work at the same rate, so the same absolute quantity is removed per minute — meaning the fraction removed halves. Since Cl = Q × ER, and doubling Q halves ER, the product is unchanged. This reciprocal relationship is the reason low-ER clearance is flow-independent, and it is the point most often missed.

Why does enzyme activity matter so much? Because the enzymes are the bottleneck. Enzyme induction directly increases the rate-limiting step, and clearance rises in proportion. Enzyme inhibition reduces it correspondingly. This is why low-ER drugs — warfarin, phenytoin, theophylline — dominate the list of clinically important interactions.

Why does protein binding matter? Because only free drug can enter the hepatocyte and reach the enzyme, and here extraction is slow enough that the bound reservoir is not meaningfully drawn upon during a single transit. The enzyme effectively sees f_u × C, so clearance is proportional to the free fraction — exactly as the limiting expression states.

Intermediate extraction ratio (0.3–0.7) — midazolam and sufentanil — are partially dependent on all three factors, which is precisely why midazolam behaves unpredictably in the critically ill.