eChapter 5. Clinical Pharmacokinetics and Pharmacodynamics

• Clinical pharmacokinetics is the discipline that describes the absorption, distribution, metabolism, and elimination of drugs in patients requiring drug therapy.
• Clearance is the most important pharmacokinetic parameter because it determines the steady-state concentration for a given dosage rate. Physiologically, clearance is determined by blood flow to the organ that metabolizes or eliminates the drug and the efficiency of the organ in extracting the drug from the bloodstream.
• The volume of distribution is a proportionality constant that relates the amount of drug in the body to the serum concentration. The volume of distribution is used to calculate the loading dose of a drug that will immediately achieve a desired steady-state concentration. The value of the volume of distribution is determined by the physiologic volume of blood and tissues and how the drug binds in blood and tissues.
• Half-life is the time required for serum concentrations to decrease by one-half after absorption and distribution are complete. It is important because it determines the time required to reach steady state and the dosage interval. Half-life is a dependent kinetic variable because its value depends on the values of clearance and volume of distribution.
• The fraction of drug absorbed into the systemic circulation after extravascular administration is defined as its bioavailability.
• Most drugs follow linear pharmacokinetics, whereby steady-state serum drug concentrations change proportionally with long-term daily dosing.
• Some drugs do not follow the rules of linear pharmacokinetics. Instead of steady-state drug concentration changing proportionally with the dose, serum concentration changes more or less than expected. These drugs follow nonlinear pharmacokinetics.
• Pharmacokinetic models are useful to describe data sets, to predict serum concentrations after several doses or different routes of administration, and to calculate pharmacokinetic constants such as clearance, volume of distribution, and half-life. The simplest case uses a single compartment to represent the entire body.
• Factors to be taken into consideration when deciding on the best drug dose for a patient include age, gender, weight, ethnic background, other concurrent disease states, and other drug therapy.
• Cytochrome P450 is a generic name for the group of enzymes that are responsible for most drug metabolism oxidation reactions. Several P450 isozymes have been identified, including CYP1A2, CYP2C9, CYP2C19, CYP2D6, CYP2E1, and CYP3A4.
• Membrane transporters are protein molecules concerned with the active transport of drugs across cell membranes. The importance of transport proteins in drug bioavailability, elimination, and distribution is continuing to evolve. A principal transport protein involved in the movement of drugs across biologic membranes is P-glycoprotein. P-glycoprotein is present in many organs, including the gastrointestinal (GI) tract, liver, and kidney. Other transport protein families include the organic cation transporters, the organic anion transporters, and the organic anion transporting polypeptides.
• When deciding on initial doses for drugs that are renally eliminated, the patient's renal function should be assessed. A common, useful way to do this is to measure the patient's serum creatinine concentration and convert this value into an estimated creatinine clearance (CLcr est). For drugs that are eliminated primarily by the kidney (≥60% of the administered dose), some agents will need minor dosage adjustments for CLcr est between 30 and 60 mL/min (0.50 and 1.00 mL/s), moderate dosage adjustments for CLcr est between 15 and 30 mL/min (0.25 and 0.50 mL/s), and major dosage adjustments for CLcr est less than 15 mL/min (0.25 mL/s). Supplemental doses of some medications also may be needed for patients receiving hemodialysis if the drug is removed by the artificial kidney or for patients receiving hemoperfusion if the drug is removed by the hemofilter.
• When deciding on initial doses for drugs that are hepatically eliminated, the patient's liver function should be assessed. The Child-Pugh score can be used as an indicator of a patient's ability to metabolize drugs that are eliminated by the liver. In the absence of specific pharmacokinetic dosing guidelines for a medication, a Child-Pugh score equal to 8 or 9 is grounds for a moderate decrease (˜25%) in the initial daily drug dose for agents that are metabolized primarily hepatically (≥60%), and a score of 10 or greater indicates that a significant decrease in the initial daily dose (˜50%) is required for drugs that are metabolized mostly hepatically.
• For drugs that exhibit linear pharmacokinetics, steady-state drug concentration (Css) changes proportionally with dose (D). To adjust a patient's drug therapy, a reasonable starting dose is administered for an estimated three to five half-lives. A serum concentration is obtained, assuming that it will reflect Css. Independent of the route of administration, the new dose (Dnew) needed to attain the desired Css(Css,new) is calculated as Dnew = Dold(Css,new/Css,old), where Dold and Css,old are the old dose and old Css, respectively.
• If it is necessary to determine the pharmacokinetic constants for a patient to individualize his or her dose, a small pharmacokinetic evaluation is conducted in the individual. Additionally, Bayesian computer programs that aid in the individualization of therapy are available for many different drugs.
• Pharmacodynamics is the study of the relationship between the concentration of a drug and the response obtained in a patient. If pharmacologic effect is plotted against concentration for most drugs, a hyperbola results with an asymptote equal to the maximum attainable effect.

On completion of the chapter, the reader will be able to:

1. Define the following terms: clinical pharmacokinetics, pharmacodynamics, clearance, volume of distribution, half-life, bioavailability, linear pharmacokinetics, and nonlinear pharmacokinetics.

2. Calculate a maintenance dose, loading dose, and dosage interval for a patient given values of clearance, volume of distribution, and half-life.

3. Compare the attributes of linear pharmacokinetics and nonlinear pharmacokinetics.

4. List patient characteristics needed to decide on the best drug dose for an individual.

5. Discuss the various drug metabolism enzymes and drug transport proteins and their importance in drug bioavailability and elimination.

6. Calculate the estimated creatinine clearance and Child-Pugh score for a patient.

7. Recommend when drug doses should be modified for patients with renal or hepatic dysfunction.

8. Calculate a modified drug dosage regimen for an agent that follows linear pharmacokinetics given a steady-state drug concentration and current drug dose.

9. Identify when a patient would benefit from the determination of pharmacokinetic constants for the use of dosage adjustment using drug-specific techniques or Bayesian computer dosing programs.

10. Calculate an initial dose for a patient for the following medications: aminoglycoside antibiotics, vancomycin, digoxin, theophylline, phenytoin, and cyclosporine.

Pharmacokinetic concepts have been used successfully by pharmacists to individualize patient drug therapy for about a quarter century. Pharmacokinetic consultant services and individual clinicians routinely provide patient-specific drug-dosing recommendations that increase the efficacy and decrease the toxicity of many medications. Laboratories routinely measure patient serum or plasma samples for many drugs, including antibiotics (e.g., aminoglycosides and vancomycin), theophylline, antiepileptics (e.g., phenytoin, carbamazepine, valproic acid, phenobarbital, and ethosuximide), methotrexate, lithium, antiarrhythmics (e.g., lidocaine and digoxin), and immunosuppressants (e.g., cyclosporine and tacrolimus). Combined with a knowledge of the disease states and conditions that influence the disposition of a particular drug, kinetic concepts can be used to modify doses to produce serum drug concentrations that result in desirable pharmacologic effects without unwanted side effects. This narrow range of concentrations within which the pharmacologic response is produced and adverse effects prevented in most patients is defined as the therapeutic range of the drug. eTable 5-1 lists the therapeutic ranges for commonly used medications.

Although most individuals experience favorable effects with serum drug concentrations in the therapeutic range, the effects of a given serum concentration can vary widely among individuals. Clinicians should never assume that a serum concentration within the therapeutic range will be safe and effective for every patient. The response to the drug, such as the number of seizures a patient experiences while taking an antiepileptic agent, should always be assessed when serum concentrations are measured.

Throughout this chapter, abbreviations for various pharmacokinetic parameters are used frequently. eTable 5-2 lists commonly used abbreviations.

Clinical pharmacokinetics is the discipline that describes the absorption, distribution, metabolism, and elimination of drugs in patients requiring drug therapy. When a drug is administered extravascularly to patients, it must be absorbed across biologic membranes to reach the systemic circulation. If the drug is given orally, the drug molecules must pass through the gastrointestinal (GI) tract wall into capillaries. For transdermal patches, the drug must penetrate the skin to enter the vascular system. In general, the pharmacologic effect of the drug is delayed when it is given extravascularly because time is required for the drug to be absorbed into the vascular system.

The vascular system generally provides the “transportation” for the drug molecule to its site of activity. After the drug reaches the systemic circulation, it can leave the vasculature and penetrate the various tissues or remain in the blood. If the drug remains in the blood, it may bind to endogenous protein, such as albumin and α1-acid glycoprotein. This binding usually is reversible, and an equilibrium is created between protein-bound drug and unbound drug. Unbound drug in the blood provides the driving force for distribution of the agent to body tissues. If unbound drug leaves the bloodstream and distributes to tissue, it may become tissue-bound, it may remain unbound in the tissue, or if the tissue can metabolize or eliminate the drug, it may be rendered inactive and/or eliminated from the body. If the drug becomes tissue-bound, it may bind to the receptor that causes its pharmacologic or toxic effect or to a nonspecific binding site that causes no effect. Again, tissue binding is usually reversible, so that the tissue-bound drug is in equilibrium with the unbound drug in the tissue.

Certain organs—such as the liver, GI tract wall, and lung—possess enzymes that metabolize drugs. The resulting metabolite may be inactive or have a pharmacologic effect of its own. The blood also contains esterases, which cleave ester bonds in drug molecules and generally render them inactive.

Drug metabolism usually occurs in the liver through one or both of two types of reactions. Phase I reactions generally make the drug molecule more polar and water soluble so that it is prone to elimination by the kidney. Phase I modifications include oxidation, hydrolysis, and reduction. Phase II reactions involve conjugation to form glucuronides, acetates, or sulfates. These reactions generally inactivate the pharmacologic activity of the drug and may make it more prone to elimination by the kidney.

Other organs have the ability to eliminate drugs or metabolites from the body. The kidney can excrete drugs by glomerular filtration or by such active processes as proximal tubular secretion. Drugs also can be eliminated via bile produced by the liver or air expired by the lungs.

Linear Pharmacokinetics

Most drugs follow linear pharmacokinetics; serum drug concentrations change proportionally with long-term daily dosing. For example, if a drug dose were doubled from 300 to 600 mg daily, the patient's serum drug concentration would double.

When a drug is given by continuous IV infusion, serum concentrations increase until an equilibrium is established between the drug dosage rate and the rate of drug elimination. At that point, the rate of drug administration equals the rate of drug elimination, and the serum concentrations remain constant (eFig. 5-1). For example, if a patient were receiving a continuous IV infusion of theophylline at 40 mg/h, the theophylline serum concentration would increase until the patient's body was eliminating theophylline at 40 mg/h. When serum drug concentrations reach a constant value, steady state is achieved.

If the drug is given at intermittent dosage intervals, such as 250 mg every 6 hours, steady state is achieved when the serum-concentration-versus-time curves for each dosage interval are superimposable. The amount of drug eliminated during the dosage interval equals the dose.

Bioavailability and Bioequivalence

When drugs are administered extravascularly, drug molecules must be released from the dosage form (dissolution) and pass through several biologic barriers before reaching the vascular system (absorption). The fraction of drug absorbed into the systemic circulation (F) after extravascular administration is defined as its bioavailability and can be calculated after single IV and extravascular doses as1

where D and Div are the extravascular and IV doses, respectively, and AUCiv,0-∞ and AUC0-∞ are the IV and extravascular areas under the serum- or blood-concentration-versus-time curves, respectively, from time zero to infinity. The AUC represents the body's total exposure to the drug and is a function of the fraction of the drug dose that enters the systemic circulation via the administered route and clearance (eFig. 5-2). When F is less than 1 for a drug administered extravascularly, either the dosage form did not release all the drug contained in it, or some of the drug was eliminated or destroyed (by stomach acid or other means) before it reached the systemic circulation.

When the extravascular dose is administered orally, part of the dose may be metabolized by enzymes or removed by transport proteins contained in the GI tract wall or liver before it reaches the systemic circulation.2, 3 This occurs commonly when drugs have a high liver extraction ratio or are subject to GI tract wall metabolism because, after oral administration, the drug must pass through the GI tract wall and into the portal circulation of the liver. Transport proteins are also present in the GI tract wall that can actively pump drug molecules that already have been absorbed back into the lumen of the GI tract. P-glycoprotein (P-gp) is the primary transport protein that interferes with drug absorption by this mechanism. For example, if an orally administered drug is 100% absorbed from the GI tract but has a hepatic extraction ratio of 0.75, only 25% of the original dose enters the systemic circulation. This first-pass effect through the liver and/or GI tract wall is avoided when the drug is given by other routes of administration. The computation of F does not separate loss of oral drug metabolized by the first-pass effect and drug not absorbed by the GI tract. Special techniques are needed to determine the fraction of drug absorbed orally for drugs with high liver extraction ratios or substantial gut wall metabolism.

Two different dosage forms of the same drug are considered to be bioequivalent when the AUC0-∞, maximum serum or blood concentrations (Cmax), and the times that Cmax occurs (tmax) are neither clinically nor statistically different. When this occurs, the serum-concentration-versus-time curves for the two dosage forms should be superimposable and identical. Bioequivalence studies have become very important as expensive drugs become available in less costly generic form. Most bioequivalence studies involve 18 to 25 healthy adults who are given the brand-name product and the generic product in a randomized, crossover study design.

Clearance

Clearance (CL) is the most important pharmacokinetic parameter because it determines the steady-state drug concentration (Css) for a given dosage rate. When a drug is given at a continuous IV infusion rate equal to k0, the Css is determined by the quotient of k0 and CL (Css = k0/CL). If the drug is administered as individual doses (D) at a given dosage interval (τ), the average Css over the dosage interval is given by the equation4

where F is the fraction of dose absorbed into the systemic vascular system. The average Css over the dosage interval is the Css that would have occurred had the same dose been given as a continuous IV infusion (e.g., 300 mg every 6 hours would produce an average Css equivalent to the actual Css produced by a continuous infusion administered at a rate of 50 mg/h).

Physiologically, clearance is determined by (a) blood flow (Q) to the organ that metabolizes (liver) or eliminates (kidney) the drug and (b) the efficiency of the organ in extracting the drug from the bloodstream.5 Efficiency is measured using an extraction ratio (E), calculated by subtracting the concentration in the blood leaving the extracting organ (Cout) from the concentration in the blood entering the organ (Cin) and then dividing the result by Cin:

Clearance for that organ is calculated by taking the product of Q and E (CL = QE). For example, if liver blood flow equals 1.5 L/min, and the drug's extraction ratio is 0.33, hepatic clearance equals 0.5 L/min. Total clearance is computed by summing all the individual organ clearance values. Clearance changes occur in patients when the blood flow to extracting organs changes or when the extraction ratio changes. Vasodilators such as hydralazine and nifedipine increase liver blood flow, whereas chronic heart failure (CHF) and hypotension can decrease hepatic blood flow. Extraction ratios can increase when enzyme inducers increase the amount of drug-metabolizing enzyme. Extraction ratios may decrease if enzyme inhibitors inhibit drug-metabolizing enzymes or necrosis causes loss of parenchyma.

Intrinsic Clearance

The extraction ratio also can be thought of in terms of the unbound fraction of drug in the blood (fb), the intrinsic ability of the extracting organ to clear unbound drug from the blood (CLint), and blood flow to the organ (Q):6, 7

By substituting this equation for E, the clearance equation becomes

Clearance changes will occur when blood flow to the clearing organ changes (in conditions where blood flow is reduced, e.g., shock and CHF, or where blood flow is increased, e.g., administration of medications, such as vasodilators, and resolution of shock or CHF), binding in the blood changes (e.g., if the concentration of binding proteins is low or highly protein-bound drugs are displaced), or intrinsic clearance of unbound drug changes (e.g., when metabolizing enzymes are induced or inhibited by other drug therapy or functional organ tissue is destroyed by disease processes).

If CLint is large (enzymes have a high capacity to metabolize the drug), the product of fb and CLint is much larger than Q. When fb(CLint) is much greater than Q, the sum of Q and fb(CLint) in the denominator of the clearance equation almost equals fb(CLint):

Substituting this expression in the denominator of the clearance equation and canceling common terms leads to the following expression for drugs with a large CLint: CL ≈ Q. In this case, clearance of the drug is equal to blood flow to the organ; such drugs are called high-clearance drugs and have large extraction ratios. Propranolol, verapamil, morphine, and lidocaine are examples of high-clearance drugs. High-clearance drugs such as these typically exhibit high first-pass effects when administered orally.

If CLint is small (enzymes have a limited capacity to metabolize the drug), Q is much larger than the product of fb and CLint. When Q is much greater than fb(CLint), the sum of Q and fb(CLint) in the denominator of the clearance equation becomes almost equal to Q: Q ≈ Q + fb(CLint). Substituting this expression in the denominator of the clearance equation and canceling common terms leads to the following expression for drugs with a small CLint: CL ≈ fb(CLint). In this case, clearance of the drug is equal to the product of the fraction unbound in the blood and the intrinsic ability of the organ to clear unbound drug from the blood; such drugs are known as low-clearance drugs and have small extraction ratios. Warfarin, theophylline, diazepam, and phenobarbital are examples of low-clearance drugs.

As mentioned previously, the concentration of unbound drug in the blood is probably more important pharmacologically than the total (bound plus unbound) concentration. The unbound drug in the blood is in equilibrium with the unbound drug in the tissues and reflects the concentration of drug at its site of action. Therefore, the pharmacologic effect of a drug is thought to be a function of the concentration of unbound drug in the blood. The unbound steady-state concentration (Css,u) can be calculated by multiplying Css and fb: Css,u = Cssfb. The effect that changes in Q, fb, and CLint have on Css,u and therefore on the pharmacologic response of a drug depends on whether a high- or low-clearance drug is involved. Because CL = Q for high-clearance drugs, a change in fb or CLint does not change CL or Css(Css = k0/CL). However, a change in unbound drug fraction does alter Css,u(Css,u = fbCss), thereby affecting the pharmacologic response. Plasma protein-binding displacement drug interactions can be very important clinically, but they are also dangerous because the changes in Css,u are not reflected in changes in Css for high-clearance drugs. Because laboratories usually measure only total concentrations (concentrations of unbound drug are difficult to determine), the interaction is hard to detect. If CLint changes for high-clearance drugs, CL, Css, Css,u, and pharmacologic response do not change. Changes in Q cause a change in CL; changes in Css, Css,u, and drug response are indirectly proportional to changes in CL.

For low-clearance drugs, total clearance is determined by unbound drug fraction and intrinsic clearance: CL = fb(CLint). A change in Q does not change CL, Css, Css,u, or pharmacologic response. However, a change in fb or CLint does alter CL and Css (Css = k0/CL). Changes in CLint will cause a proportional change in CL.

Changes in Css, Css,u, and drug response are indirectly proportional to changes in CL. Altering fb for low-clearance drugs produces interesting results. A change in fb alters CL and Css (Css = k0/CL). Because CL and Css change in opposite directions with changes in fb, Css,u (Css,u = fbCss) and pharmacologic response do not change with alterations in the fraction of unbound drug in the blood. For example, a low-clearance drug is administered to a patient until steady-state is achieved:

Suppose that another drug is administered to the patient that displaces the first drug from plasma-protein-binding sites and doubles fb (fb now equals 2fb). CL doubles because of the protein-binding displacement [2CL = 2fb(CLint)], and Css decreases by one-half because of the change in clearance [½(Css) = k0/(2Cl)]. Css,u does not change because even though fb is doubled, Css decreased by one-half (Css,u = fbCss). The potential for error in this situation is that clinicians may increase the dose of a low-clearance drug after a protein-binding displacement interaction because Css decreased. Because Css,u and the pharmacologic effect do not change, the dose should remain unaltered. Plasma protein binding decreases occur commonly in patients taking phenytoin. Low albumin concentrations (as in trauma or pregnant patients), high concentrations of endogenous plasma protein-binding displacers (as with high concentrations of bilirubin), or plasma protein-binding drug interactions (as with concomitant therapy with valproic acid) can result in subtherapeutic total phenytoin concentrations. Despite this fact, unbound phenytoin concentrations usually are within the therapeutic range, and often the patient is responding appropriately to treatment. Thus, in these situations, unbound rather than total phenytoin serum concentrations should be monitored and used to guide future therapeutic decisions.

Clearances for Different Routes of Elimination and Metabolic Pathways

Clearances for individual organs can be computed if the excretion the organ produces can be obtained. For example, renal clearance can be calculated if urine is collected during a pharmacokinetic experiment. The patient empties his or her bladder immediately before the dose is given. Subsequent urine production is collected until the last serum concentration (Clast) is obtained. Renal clearance (CLR) is computed by dividing the amount of drug excreted in the urine by AUC0–t,last. Biliary and other clearance values are computed in a similar fashion.

Clearances also can be calculated for each metabolite that is formed from the parent drug. This computation is particularly useful in drug-interaction studies to determine which metabolic pathway is stimulated or inhibited. In the following metabolic scheme, the parent drug (D) is metabolized into two different metabolites (M1, M2) that subsequently are eliminated by the kidney (M1R, M2R):

To compute the formation clearance of M1 and M2 (CLFM1, CLFM2), urine would be collected for five or more half-lives after a single dose or during a dosage interval at steady state. The amount of metabolite eliminated in the urine is then determined. The fraction of the dose (in moles, because the molecular weights of the parent drug and metabolites are not equal) eliminated by each metabolic pathway (fM1 = M1R/D and fM2 = M2R/D) can then be computed. Formation clearance for each pathway can be calculated using the following equations: CLFM1 = fM1CLM and CLFM2 = fM2CLM, where CLM is the metabolic clearance for the parent drug.

Volume of Distribution

The volume of distribution (VD) is a proportionality constant that relates the amount of drug in the body to the serum concentration (amount in body = CVD). VD is used to calculate the loading dose (LD) of a drug that will immediately achieve a desired Css (LD = CssVD). However, in practice, the patient's own VD is not known at the time the loading dose is administered. In this case, an average VD is assumed and used to calculate a loading dose. Because the patient's VD is almost always different from the average VD for the drug, a loading dose does not attain the calculated Css, but it ideally achieves a therapeutic concentration. As usual, steady-state conditions are achieved in three to five half-lives for the drug.

The numeric value for the volume of distribution is determined by the physiologic volume of blood and tissues and how the drug binds in blood and tissues:8

where Vb and Vt are the volumes of blood and tissues, respectively, and fb and ft are the fractions of unbound drug in blood and tissues, respectively.

Half-Life

Half-life (t1/2) is the time required for serum concentrations to decrease by one-half after absorption and distribution are complete. It takes the same amount of time for serum concentrations to drop from 200 to 100 mg/L as it does for concentrations to decline from 2 to 1 mg/L (eFig. 5-3).

Half-life is important because it determines the time required to reach steady state and the dosage interval. It takes approximately three to five half-lives to reach steady-state concentrations during continuous dosing. In three half-lives, serum concentrations are at ˜90% of their ultimate steady-state values. Because most serum drug assays have an ˜10% error, it is difficult to differentiate concentrations that are within 10% of each other. For this reason, many clinicians consider concentrations obtained after three half-lives to be Css.

Half-life is also used to determine the dosage interval for a drug. For example, it may be desirable to maintain maximum steady-state concentrations at 20 mg/L and minimum steady-state concentrations at 10 mg/L. In this case, it would be necessary to administer the drug every half-life because the minimum desirable concentration is one-half the maximum desirable concentration.

Half-life is a dependent kinetic variable because its value depends on the values of CL and VD.8 The equation that describes the relationship among the three variables is t1/2 = 0.693VD/CL. Changes in t1/2 can result from a change in either VD or CL; a change in t1/2 does not necessarily indicate that CL has changed. Half-life can change solely because of changes in VD. The elimination rate constant (k) is related to the half-life by the following equation: k = 0.693/t1/2. Both the half-life and elimination rate constant describe how quickly serum concentrations decrease in the serum or blood.

Nonlinear Pharmacokinetics

Michaelis-Menten Kinetics

Some drugs do not follow the rules of linear pharmacokinetics. Instead of Css and AUC increasing proportionally with dose, serum concentrations change more or less than expected (eFig. 5-4). One explanation for the greater-than-expected increase in Css and AUC after an increase in dose is that the enzymes responsible for the metabolism or elimination of the drug may start to become saturated. When this occurs, the maximum rate of metabolism (Vmax) for the drug is approached. This is called Michaelis-Menten kinetics. The serum concentration at which the rate of metabolism equals Vmax/2 is Km. Practically speaking, Km is the serum concentration at which nonproportional changes in Css and AUC start to occur when the dose is increased. The Michaelis-Menten constants (Vmax and Km) determine the dosage rate (DR) needed to maintain a given Css: DR = VmaxCss/(Km + Css). Most drugs eliminated by the liver are metabolized by enzymes but still appear to follow linear kinetics. The reason for this disparity is that the therapeutic range for most drugs is well below the Km of the enzyme system that metabolizes the agent. The therapeutic range is higher than Km for some commonly used drugs. The average Km for phenytoin is about 4 mg/L (16 μmol/L). The therapeutic range for phenytoin is usually 10 to 20 mg/L (40 to 79 μmol/L). Most patients experience Michaelis-Menten kinetics while taking phenytoin.

Nonlinear Protein Binding

Another type of nonlinear kinetics can occur if Css and AUC increase less than expected after an increase in dose of a low-clearance drug. This usually indicates that plasma protein-binding sites are starting to become saturated, so that fb increases with increases in the dose (see eFig. 5-4). For a low-clearance drug, CL depends on the values of fb and CLint (CL = fbCLint). When a dosage increase takes place, fb increases because nearly all plasma protein-binding sites are occupied, and no binding sites are available. If fb increases, CL increases, and Css increases less than expected with the dosage change (Css = k0/CL). However, Css,u increases proportionally with the dose because Css,u depends on CLint for low-clearance drugs (Css,u = k0/CLint). Valproic acid9 and disopyramide10 both follow saturable protein-binding pharmacokinetics.

Autoinduction

For some drugs, clearance increases as the dose or concentration of the drug increases. In this situation, increasing the drug dose or concentration increases the ability of the enzyme system to eliminate the compound and to clear the drug from the body. This is usually accomplished by inducing the enzyme system responsible for the metabolism of the drug, so that the intrinsic clearance of the drug increases. Because the drug itself is causing the induction effect, this process is called autoinduction. For some drugs, such as carbamazepine,11 the autoinduction effect is continuous within the typical dosage range, which produces a curve for the dose versus Css or AUC plot similar to nonlinear protein binding (see eFig. 5-4). Detailed pharmacokinetic studies are conducted to differentiate between nonlinear protein binding and autoinduction when dose versus Css or AUC plots systematically deviate below the linear line.

Pharmacokinetic Models and Equations

Pharmacokinetic models are useful to describe data sets, to predict serum concentrations after several doses or different routes of administration, and to calculate pharmacokinetic constants such as CL, VD, and t1/2.12 Compartmental models depict the body as one or more discrete compartments to which a drug is distributed and/or from which a drug is eliminated. The shape of the serum-concentration-versus-time curve determines the number of compartments in the pharmacokinetic model and the equation used in computations (eFig. 5-5). First-order rate constants, known as microconstants, describe the rate of transfer from one compartment to another. Each compartment also has its own VD. For clinical dosage adjustment purposes using drug concentrations, a one-compartment model is the most commonly used pharmacokinetic model.

One-Compartment Model

The simplest case uses a single compartment to represent the entire body (see eFig. 5-5). The drug enters the compartment by continuous IV infusion (k0), absorption from an extravascular site with an absorption rate constant of ka, or IV bolus (D). After an IV bolus, serum concentrations decline in a straight line when plotted on semilogarithmic coordinates (see eFig. 5-3). The slope of the line is −k/2.303; t1/2 can be computed by determining the time required for concentrations to decrease by one-half (t1/2 = 0.693/k). The equation that describes the data is C = (D/VD)e−kt. VD is calculated by dividing the IV dose by the y intercept (the concentration at time zero, C0) of the graph. CL is computed by taking the product of k and VD. Once VD and k are known, concentrations at any time after the dose can be computed [C = (D/VD)e−kt].

When an extravascular dose is given, one-compartment-model serum concentrations rise during absorption, reach Cmax, then decrease in a straight line with a slope equal to –k/2.303. The equation that describes the data is C = {(FDka)/[VD(ka – k)]}(e−kt – e−kat), where F is the fraction of the dose absorbed into the systemic circulation. The absorption rate constant (ka) is obtained using the method of residuals.

The method of residuals is used to obtain the individual rate constants (eFig. 5-6). A is determined by extrapolating the terminal slope to the y axis; k can be obtained by calculating the slope or t1/2 and using the formulas given for the IV bolus case. At each time point in the absorption portion of the curve, the concentration value from the extrapolated line is noted and called the extrapolated concentration. For each point, the actual concentration is subtracted from the extrapolated concentration to compute the residual concentration. When the residual concentrations are plotted on semilogarithmic coordinates (see eFig. 5-6, inset), a line with y intercept equal to A and slope equal to −ka/2.303 is obtained. When these values are calculated, they can be placed into the equation (C = Ae−kt − Ae−kat, where A = FDka/[VD(ka − k)]) and used to compute the serum concentration at any time after the extravascular dose. The intercepts and rate constants also can be used to compute CL and VD: CL = FD/(A/k − A/ka) and VD = CL/k, where F is the fraction of the dose absorbed into the systemic circulation.

During a continuous IV infusion, the serum concentrations in a one-compartment model change according to the following function: C = (k0/CL)(1 − e−kt). If the infusion has been running for more than three to five half-lives, the patient will be at steady state, and CL can be calculated (CL = k0/Css). When the infusion is discontinued, serum concentrations appear to decline in a straight line when plotted on semilogarithmic paper with a slope of −k/2.303. VD is computed by dividing CL by k (eFig. 5-7).

Multicompartment Model

After an IV bolus dose, serum concentrations often decline in two or more phases. During the early phases, the drug leaves the bloodstream by two mechanisms: (a) distribution into tissues and (b) metabolism and/or elimination. Because the drug is leaving the bloodstream through these two mechanisms, serum concentrations decline rapidly. After tissues and blood are in equilibrium, only metabolism and elimination remove the drug from the blood. During this terminal phase, serum concentrations decline more slowly. The half-life is measured during the terminal phase by determining the time required for concentrations to decline by one-half.

After an IV bolus dose, serum concentrations decrease as if the drug were being injected into a central compartment that not only metabolizes and eliminates the drug but also distributes the drug to one or more other compartments. Of these multicompartment models, the two-compartment model is encountered most commonly (see eFig. 5-5). After an IV bolus injection, serum concentrations decrease in two distinct phases, described by the equation

or C = Ae−αt + Be−βt, where k21 is the first-order rate constant that reflects the transfer of the drug from compartment 2 to compartment 1, VD1 is the VD of compartment 1, A = D(α − k21)/[VD1(α − β)] and B = D(k21 − β)/[VD1(α − β)]. The rate constants α and β found in the exponents of the equations describe the distribution and elimination of the drug, respectively (eFig. 5-8). A and B are the y intercepts of the lines that describe drug distribution and elimination, respectively, on the log concentration-versus-time plot.

The residual line is calculated as before using the method of residuals (see eFig. 5-8, inset). The terminal line is extrapolated to the y axis, and extrapolated concentrations are determined for each time point. Because actual concentrations are greater in this case, residual concentrations are calculated by subtracting the extrapolated concentrations from the actual concentrations. When plotted on semilogarithmic paper, the residual line has a y intercept equal to A. The slope of the residual line is used to compute α (slope = −α/2.303). With the rate constants (α and β) and the intercepts (A and B), concentrations can be calculated for any time after the IV bolus dose (C = Ae−αt + Be−βt), or pharmacokinetic constants can be computed: CL = D/[(A/α) + (B/β)], VD,β = CL/β, VD,ss = {D[(A/α2) + (B/β2)]}/[(A/α) + (B/β)]2.

If serum concentrations of a drug given as a continuous IV infusion decline in a biphasic manner after the infusion is discontinued, a two-compartment model describes the data set (eFig. 5-9).13, 14 In this instance, the postinfusion concentrations decrease according to the equation C = Re−αt′ + Se−βt′, where t′ is the postinfusion time (t' = 0 when infusion is discontinued), and R, S, α, and β are determined from the postinfusion concentrations using the method of residuals with the y axis set at t′ = 0. R and S are used to compute A and B. A and B are the y intercepts that would have occurred had the total dose given during the infusion (D = k0T) been administered as an IV bolus dose:

where T is the duration of infusion. Once A, B, α, and β are known, the equations for an IV bolus are used to compute the pharmacokinetic constants. Often, when a drug is given as an IV bolus or continuous IV infusion, a two-compartment model is used to describe the data, but when the same agent is given extravascularly, a one-compartment model applies.15 In this case, distribution occurs during the absorption phase, so a distribution phase is not observed.

Volumes of Distribution in Multicompartment Models

Two different VD values are needed as proportionality constants for drugs that require multicompartment models to describe the serum-concentration-versus-time curve. The VD that is used to compute the amount of drug in the body during the terminal (β) portion of the curve is called VD,β (amount of drug in body = VD,β C). During a continuous IV infusion at steady state, VD,ss is used to compute the amount of drug in the body (amount of drug in body = VD,ssC). VD,ss is also the VD that can be computed using the physiologic volumes of blood and tissues and the ratio of unbound drug in blood to that in tissues [VD,ss = Vb + (fb/ft)Vt]. Because the value of VD,β changes when CL changes, VD,ss should be used to indicate if drug distribution changes during pharmacokinetic or drug-interaction experiments.

Multiple Dosing and Steady-State Equations

Any of these compartmental equations can be used to determine serum concentrations after multiple doses. The multiple-dosing factor (1 − e−nKτ)/(1 − e−Kτ), where n is the number of doses, K is the appropriate rate constant, and τ is the dosage interval, is simply multiplied by each exponential term in the equation, substituting the rate constant of each exponent for K. Time (t) is set at 0 at the beginning of each dosage interval. For example, a single-dose two-compartment IV bolus is calculated as follows: C = Ae−αt + Be−βt. Thus, the equation for a multiple-dose two-compartment IV bolus is

A single-dose one-compartment IV bolus is calculated as C = (D/VD)e−kt. For a multiple-dose one-compartment IV bolus, the concentration is C = (D/VD)e−kt[(1 − e−nkτ)/(1 − e−kτ)].

At steady state, the number of doses becomes large, e−nKτ approaches zero, and the multiple-dosing factor equals 1/(1 − e−Kτ). Therefore, the steady-state versions of the equations are simpler than their multiple-dose counterparts:

and

for a steady-state two-compartment IV bolus and a steady-state one-compartment IV bolus, respectively.

Use of Pharmacokinetic Concepts for Individualization of Drug Therapy

Many factors must be taken into consideration when deciding on the best drug dose for a patient. For example, the age of the patient is important because the dose (in milligrams per kilogram) for pediatric patients may be higher and for geriatric patients may be lower than the typically prescribed dose for young adults. Gender also can be a factor because male and female patients metabolize and eliminate some drugs differently. Patients who are significantly obese or cachectic also may require different drug doses because of clearance and volume of distribution changes. Other drug therapy that could cause drug interactions needs to be considered. Disease states and conditions may alter the drug-dosage regimen for a patient. Three disease states that deserve special mention are CHF, renal disease, and hepatic disease. Renal and hepatic diseases cause loss of organ function and decreased drug elimination and metabolism. CHF causes decreased blood flow to organs that clear the drug from the body.

Many drug compounds are racemic mixtures of stereoisomers. In most cases, one of the isomers is more pharmacologically active than the other isomer, and each isomer may exhibit different pharmacokinetic properties. Warfarin, propranolol, verapamil, and ibuprofen are all racemic mixtures of stereoisomers. Some drug interactions inhibit or increase the elimination of only one stereoisomer. The importance of the drug interaction depends on which isomer is affected. Other drugs, such as dextromethorphan, levofloxacin, and diltiazem, are composed of just one stereoisomer.

Genetics also plays a role in drug metabolism. Cytochrome P450 is a generic term for the group of enzymes that are responsible for most drug metabolism oxidation reactions. Several cytochrome P450 (CYP) isozymes have been identified that are responsible for the metabolism of many important drugs (^57520422). CYP2C19 is responsible for aromatic hydroxylation of (S)-mephenytoin, and CYP2D6 oxidizes debrisoquine.16 These subsets of the CYP enzyme family are also responsible for the metabolism of several other drugs (CYP2D6: many tricyclic antidepressants, codeine, (S)-metoprolol; CYP2C19: most proton pump inhibitors, sertraline, voriconazole). CYP2C9, CYP2C19, and CYP2D6 isozymes appear to be under genetic control. As a consequence, there are “poor metabolizers” who have a defective mutant gene for the isozyme, cannot manufacture a fully functional isozyme, and therefore cannot metabolize the drug substrate very well. “Extensive metabolizers” have the standard gene for the isozyme and metabolize the drugs normally. Poor metabolizers usually are a minority of the general population. They may achieve toxic concentrations of a drug when usual doses are prescribed for them or, if the active drug moiety is a metabolite, may fail to have any pharmacologic effect from the drug. The ethnic background of the patient can affect the likelihood that the patient will be a poor metabolizer.16 For example, the incidence of poor metabolizers for CYP2D6 is ˜5% to 10% for whites and ˜0% to 1% for Asians, whereas for CYP2C19, poor metabolizers make up ˜3% to 6% of the white population and ˜20% of the Asian population. Approximately 7% of the whites are poor metabolizers for CYP2C9 substrates.

Other cytochrome P450 isozymes have been isolated.16 CYP1A2 is the enzyme that is responsible for the demethylation of caffeine and theophylline; CYP2C9 metabolizes phenytoin, tolbutamide, losartan, and ibuprofen; some antiretroviral protease inhibitors, cyclosporine, nifedipine, lovastatin, simvastatin, and atorvastatin are metabolized by CYP3A4; and ethanol is a substrate for CYP2E1. It is important to recognize that a drug may be metabolized by more than one cytochrome P450 isozyme. Although most tricyclic antidepressants are hydroxylated by CYP2D6, N-demethylation probably is mediated by a combination of CYP2C19, CYP1A2, and CYP3A4. Acetaminophen appears to be metabolized by both CYP1A2 and CYP2E1. The 4-hydroxy metabolite of propranolol is produced by CYP2D6, but side-chain oxidation of propranolol is probably a product of CYP2C19. The CYP3A enzyme family comprises ˜90% of the drug-metabolizing enzyme present in the intestinal wall but only ˜30% of the drug-metabolizing enzyme found in the liver. The remainder of hepatic drug-metabolizing enzyme is ˜20% for the CYP2C family, ˜13% for CYP1A2, ˜7% for CYP2E1, and ˜2% for CYP2D6.

Understanding which cytochrome P450 isozyme is responsible for the metabolism of a drug is extraordinarily useful in predicting and understanding drug interactions. Some drug-metabolism inhibitors and inducers are highly selective for certain CYP isozymes.16 Quinidine is an extremely potent inhibitor of the CYP2D6 enzyme system;16 a single 50-mg dose of quinidine can change a rapid metabolizer of debrisoquine into a poor metabolizer. Ciprofloxacin and zileuton inhibit, whereas tobacco and marijuana smoke induces, CYP1A2. Some drugs that are enzyme inhibitors are also substrates for that same enzyme system and appear to cause drug interactions by being a competitive inhibitor. For example, erythromycin is both a substrate for and an inhibitor of CYP3A4. Obviously, if one knows that a new drug is metabolized by a given CYP enzyme system, it is logical to assume that the new drug will exhibit drug interactions with the known inducers and inhibitors of that CYP isozyme.

The importance of membrane transport proteins in drug bioavailability, elimination, and distribution is now better understood.16–18 Membrane transporters are protein molecules concerned with the active transport of drugs across cell membranes (^57520426). This results in the transfer of drug molecules either out of or into cells. Membrane transporters have been found in the intestine, liver, kidney, and the blood-brain barrier.

A principal transport protein involved in the movement of drugs across biologic membranes is P-gp. P-gp is present in many organs, including the GI tract, liver, and kidney. If a drug is a substrate for P-gp, its oral absorption may be decreased when P-gp transports drug molecules that have been absorbed back into the GI tract lumen. In the liver, some drugs are transported by P-gp from the blood into the bile, where the drug is eliminated by biliary secretion. Similarly, some drugs eliminated by the kidney are transported from the blood into the urine by P-gp. Digoxin is a substrate of P-gp. Other possible mechanisms for drug interactions are when two drugs that are substrates for P-gp compete for transport by the protein and when a drug is an inhibitor or inducer of P-gp. Drug interactions involving inhibition of P-gp decrease drug transportation in these organs and potentially can increase GI absorption of an orally administered drug, decrease biliary secretion of the drug, or decrease renal elimination of drug molecules. The drug interaction between amiodarone and digoxin probably involves all three of these mechanisms; this explains why digoxin concentrations increase so dramatically in patients receiving amiodarone. Many drugs that are metabolized by CYP3A4 are also substrates for P-gp, and some of the drug interactions attributed to inhibition of CYP3A4 may be a result of decreased drug transportation by P-gp. Drug interactions involving induction of P-gp have the opposite effect in these organs and may decrease GI absorption of an orally administered drug, increase biliary secretion of the drug, or increase renal elimination of drug molecules.

Other membrane transporter families include the organic cation transporters (OCT family), organic anion transporters (OAT family), and the organic anion transporting polypeptides (OATP family).

Selection of Initial Drug Doses

When deciding on initial doses for drugs that are eliminated renally, the patient's renal function should be assessed. A common, useful way to do this is to measure the patient's serum creatinine concentration and convert this value into a CLcr est. Serum creatinine values alone should not be used to assess renal function because they do not include the effects of age, body weight, or gender. The Cockcroft-Gault equation19 is probably the most widely used method to estimate creatinine clearance (CLcr) (in milliliters per minute) in adults (age 18 years or older) who are within ˜30% of their ideal body weight and have stable renal function:

where BW is body weight (in kilograms), age is the patient's age (in years), 0.85 is a correction factor to account for lower muscle mass in women, and Scr is serum creatinine (in milligrams per deciliter). For children, the following estimation equations are available according to the age of the child20 age 0 to 1 year: CLcr est (in mL/min/1.73 m2) = (0.45 × Lt)/Scr; age 1 to 20 years: CLcr est (in mL/min/1.73 m2) = (0.55 × Lt)/Scr, where Lt is patient length in centimeters. (To use these equations, Scr expressed in μmol/L must first be divided by 88.4 to obtain conventional units of mg/dL. Conversion of CLcr to units of mL/s/m2 requires multiplication of CLcr expressed in milliliters per minute per 1.73 m2 by 0.00963.) Other methods to determine CLcr est for obese adults21 and patients with rapidly changing renal function22 are available. Creatinine is a by-product of muscle breakdown in the body, so none of these estimation methods work well in patients with muscle disease, such as multiple sclerosis, or diseases that alter muscle mass, such as cachexia, malnutrition, cancer, or spinal cord injury. Nomograms that adjust initial doses according to a patient's renal function are available for several drugs, including digoxin,23 vancomycin,24 and the aminoglycoside antibiotics.25

For drugs that are eliminated primarily by the kidney (≥60% of the administered dose), some agents will need minor dosage adjustments for CLcr est between 30 and 60 mL/min (0.50 and 1.00 mL/s), moderate dosage adjustments for CLcr est between 15 and 30 mL/min (0.25 and 0.50 mL/s), and major dosage adjustments for CLcr est less than 15 mL/min (0.25 mL/s). Specific recommendations for dosage adjustments of other drugs for patients with renal disease are available.26, 27 Supplemental doses of some medications also may be needed for patients receiving hemodialysis if the drug is removed by the artificial kidney or for patients receiving hemoperfusion if the drug is removed by the hemofilter.27

A similar assessment of liver function should be made for drugs that are metabolized hepatically. Unfortunately, there is no single test that can estimate liver drug-metabolism capacity accurately, and those that are used do not always prove accurate. High aminotransferase (aspartate aminotransferase [AST] and alanine aminotransferase [ALT]) and alkaline phosphatase concentrations usually indicate acute hepatic cellular damage and do not establish poor liver drug metabolism reliably. Abnormal values for three tests that usually indicate that drugs will be metabolized poorly by the liver are high serum bilirubin concentration, low serum albumin concentration, and a prolonged prothrombin time. Bilirubin is metabolized by the liver, and albumin and clotting factors are manufactured by the liver, so aberrant values for all three of these tests are a more reliable indicator of abnormal liver drug metabolism. The Child-Pugh score,28 a widely used clinical classification for liver disease that incorporates clinical signs and symptoms (ascites and hepatic encephalopathy), in addition to these three laboratory tests, can be used as an indicator of a patient's ability to metabolize drugs that are eliminated by the liver. A score in excess of 10 suggests very poor liver function. As a general rule, patients with cirrhosis have the most severe decreases in liver drug metabolism. Patients with acute or chronic hepatitis often retain relatively normal or slightly decreased hepatic drug-metabolism capacity. In the absence of specific pharmacokinetic dosing guidelines for a medication, a Child-Pugh score equal to 8 to 9 is grounds for a moderate decrease (˜25%) in initial daily drug dose for agents that are metabolized primarily (≥60%) hepatically, and a score of 10 or greater indicates that a significant decrease in initial daily dose (˜50%) is required for drugs that are metabolized mostly by the liver. As in any patient with or without liver dysfunction, initial doses are meant as starting points for dosage titration based on patient response and avoidance of adverse effects.

Because there are no good markers of liver function, clinicians have come to rely on pharmacokinetic parameters derived in various patient populations to compute initial doses of drugs that are eliminated hepatically. eTable 5-4 contains average pharmacokinetic parameters for theophylline in several disease states. Initial doses of many liver-metabolized drugs are computed by determining which disease states and/or conditions the patient has that are known to alter the kinetics of the drug and by using these average pharmacokinetic constants to calculate doses. The patient is then monitored for therapeutic and adverse effects, and drug serum concentrations are obtained to ensure that concentrations are appropriate and to adjust doses, if necessary. The following computations illustrate the estimated IV loading dose and the IV continuous infusion necessary to achieve a theophylline concentration of 10 mg/L (10 mcg/mL; 55.5 μmol/L) for a 55-year-old man, weighing 70 kg (154 lb), with liver cirrhosis (mean kinetic parameters obtained from eTable 5-4):

If theophylline is to be given as the aminophylline salt form, each dose would need to be changed to reflect the fact that aminophylline contains only 85% theophylline (LD = 350 mg of theophylline/0.85 = 410 mg of aminophylline infused over 20 to 30 minutes, k0 = 15 mg/h of theophylline/0.85 = 18 mg/h of aminophylline to begin after loading dose is given).

Heart failure is often overlooked as a disease state that can alter drug disposition. Severe heart failure decreases cardiac output and therefore reduces liver blood flow. Theophylline,29 lidocaine,30 and drugs with high extraction ratios are compounds whose clearance declines with decreased liver blood flow. Initial dosages of these drugs should be reduced in patients with moderate to severe heart failure (New York Heart Association class III or IV) by 25% to 50% until steady-state concentrations and response can be determined.

Use of Steady-State Drug Concentrations

Serum drug concentrations are readily available to clinicians to use as guides for the individualization of drug therapy. The therapeutic ranges for several drugs have been identified, and it is likely that new drugs also will be monitored using serum concentrations. Although several individualization methods have been advocated for specific drugs, one simple, reliable method is used commonly. For drugs that exhibit linear pharmacokinetics, Css changes proportionally with the dose. To adjust a patient's drug therapy, a reasonable starting dose is administered for an estimated three to five half-lives. A serum concentration is obtained, assuming that it will reflect Css. Independent of the route of administration, the new dose (Dnew) needed to attain the desired Css (Css,new) is calculated: Dnew = Dold(Css,new/Css,old), where Dold and Css,old are the old dose and old Css, respectively. To use this method, Css,old must reflect steady-state conditions. Often patients are noncompliant with regard to their drug dosage and therefore are not at steady state. This occurs not only in outpatients but also in hospital inpatients. Inpatients can spit out oral doses or alter the infusion rates on IV pump rates after the nurse leaves the hospital room. Doses also can be missed if the patient is absent from his or her room at the time medications are to be administered. If Css,old is much larger or smaller than expected for the Dold the patient is taking, one should suspect noncompliance and repeat the serum concentration determination after another three to five half-lives or change the patient's dose cautiously and monitor for signs of toxicity or lack of effect.

Measurement of Pharmacokinetic Parameters in Patients

If it is necessary to determine the kinetic constants for a patient to individualize his or her dose, a small kinetic evaluation is conducted in the individual. In these cases, the number of serum concentrations obtained from the patient is held to the minimum needed to calculate accurate pharmacokinetic parameters and doses. The reason for using fewer serum drug concentration determinations is to be as cost-effective as possible because these laboratory tests generally cost $20 to $50 each.

Although many drugs follow two-compartment-model pharmacokinetics (especially after IV administration), a one-compartment model is used to compute kinetic parameters in patients because too many serum concentration determinations would be needed to determine accurately both the distribution and elimination phases found in the two-compartment model. Because of this, serum concentrations usually are not measured in patients during the distribution phase. Another important reason serum concentrations are not measured during the distribution phase for therapeutic drug-monitoring purposes in patients is that drug in the blood and drug in the tissues are not in equilibrium during this time, so that serum concentrations do not reflect tissue concentrations. When drug serum concentrations are obtained in patients for the purpose of assessing efficacy or toxicity, it is important that they be measured in the postdistribution phase when drug in the blood is in equilibrium with drug at the site of action.

In the case where the patient has received enough doses to be at steady state, pharmacokinetic parameters can be computed using a predose minimum concentration and a postdose maximum concentration. Under steady-state conditions, serum concentrations after each dose are identical, so the predose minimum concentration is the same before each dose (eFig. 5-10). This situation allows the predose concentration to be used to compute both the patient's t1/2 and V, where V = Dose/(Cmax,ss − Cmin,ss). If the drug was given extravascularly or has a significant distribution phase, the postdose concentration should be determined after absorption or distribution is finished. To ensure that steady-state conditions have been achieved, the patient needs to receive the drug on schedule for at least three to five estimated half-lives. To make sure that this is the case, inpatients should have their medication administration records checked, and the patient's nurse should be consulted regarding missed or late doses. Outpatients should be interviewed about compliance with the prescribed dosage regimen. When compliance with the dosage regimen has been verified, steady-state conditions reasonably can be assumed.

If the patient is not at steady state, an additional postdose serum concentration determination should be done to compute the patient's pharmacokinetic parameters. Ideally, the third concentration (C3) should be acquired approximately one estimated half-life after the postdose maximum concentration. Determining serum concentrations too close together will hamper the drug assay's ability to measure differences between them, and getting the third sample too late could result in a concentration too low for the assay to detect. In this situation, the predose minimum and postdose maximum concentrations are used to compute V, where V = Dose/(Cmax − Cmin), and both postdose concentrations are used to calculate t1/2 (eFig. 5-11).

After CL, V, and t1/2 have been computed for a patient, the dose and dosage interval necessary to achieve desired steady-state serum concentrations can be calculated using one-compartment-model equations. Specific examples of these methods to calculate initial doses and individualized doses using serum concentrations are discussed later in this chapter for the aminoglycoside antibiotics, vancomycin, digoxin, theophylline, phenytoin, and cyclosporine.

Computer Programs

Computer programs that aid in the individualization of therapy are available for many different drugs. The most sophisticated programs use nonlinear regression to fit CL and VD to actual serum concentrations obtained in a patient.31 After drug doses and serum concentrations are entered into the computer, nonlinear least-squares regression programs adjust CL and VD until the sum of the squared error between actual (Cact) and computer-estimated concentrations (Cest) is at a minimum [Σ(Cest − Cact)2]. Once estimates of CL and VD are available, doses are calculated easily.

Many programs also take into account what the CL and VD should be on the basis of disease states and conditions present in the patient.32 Incorporation of expected population-based parameters allows the computer to use a limited number of serum concentrations (one or two) to provide estimates of CL and VD. This type of computer program is called Bayesian because it incorporates portions of Bayes' theorem during the fitting routine.33 Bayesian pharmacokinetic dosing programs are used widely to adjust the dose of a variety of drugs. In the case of renally eliminated drugs (e.g., aminoglycosides, vancomycin, and digoxin), population estimates for kinetic parameters are generated by entering the patient's age, weight, height, gender, and serum creatinine concentration into the computer program. For hepatically eliminated drugs (e.g., theophylline and phenytoin), population estimates for kinetic parameters are computed using the patient's age, weight, and gender, as well as other factors that might change hepatic clearance, such as the presence or absence of disease states (e.g., cirrhosis or CHF) or other drug therapy that might cause a drug interaction. The population-based estimates of the pharmacokinetic parameters are then modified using nonlinear least-squares regression fits of serum concentrations to result in individualized parameters for the patient. The individualized parameters are used to compute doses for the patient that will result in desired steady-state concentrations of the drug.

Aminoglycosides

Although aminoglycoside pharmacokinetics follow multicompartment models,34 a one-compartment model appears sufficient to individualize doses in patients.35 Aminoglycosides usually are given as short-term intermittent IV infusions and administered as a single daily dose or multiple doses per day. Initial doses for aminoglycosides can be computed using estimated kinetic parameters derived from population pharmacokinetic data. The elimination rate constant is estimated using the patient's creatinine clearance in the following formula: k (in h−1) = 0.00293(CLcr) + 0.014, where CLcr is the measured or estimated creatinine clearance in milliliters per minute. The volume of distribution is estimated using the average population value for normal-weight (within 30% of ideal weight) individuals equal to 0.26 L/kg [V = 0.26(Wt), where Wt is the patient's weight] or for obese individuals (>30% of ideal weight)36 by taking into account the patient's excess adipose tissue: V = 0.26[IBW + 0.4(TBW − IBW)], where TBW is total body weight, IBW is ideal body weight [IBWmales (in kilograms) = 50 + 2.3(Ht − 60) or IBWfemales (in kilograms) = 45 + 2.3(Ht − 60), and Ht is the patient's height in inches] (height in cm can be converted to inches by multiplying by 0.394). Additional volume of distribution population estimates are available for other disease states and conditions, such as cystic fibrosis,37 ascites,38 and neonates.39

Appropriate Cmax,ss and Cmin,ss values are selected for the patient based on the site and severity of the infection and the sensitivity of the known or suspected pathogen, as well as avoidance of adverse effects. Optimal outcomes are usually associated with Cmax,ss/MIC ratios equal to 8 to 10, where MIC is the minimum inhibitory concentration for the bacteria causing the infection. For example, Cmax,ss values of 8 to 10 mg/L (8 to 10 mcg/mL) generally are selected for gram-negative pneumonia patients, whereas Cmin,ss values of less than 2 mg/L (2 mcg/mL; 4 μmol/L) usually are chosen to avoid aminoglycoside-induced nephrotoxicity when tobramycin and gentamicin are prescribed using conventional multiple-daily-dosing regimens. Once appropriate steady-state serum concentrations are selected, the dosage interval required to achieve those concentrations is calculated, and τ is rounded to a clinically acceptable value (e.g., 8, 12, 18, 24, 36, or 48 hours): τ = [(ln Cmax,ss − ln Cmin,ss)/k] + T. Finally, a dose is computed for the patient using the one-compartment-model intermittent IV infusion equation at steady state, and the dose is rounded off to the nearest 5 to 10 mg:

The Hull and Sarrubi aminoglycoside dosage nomogram (eTable 5-5) is based on this dosage-calculation method and includes precalculated doses and dosage intervals for a variety of creatinine clearance values.25 The nomogram assumes that VD = 0.26 L/kg and should not be used to compute doses for disease states with altered VD.

An example of this initial dosage scheme for a typical case is provided to illustrate the use of the various equations. Mr. JJ is a 65-year-old, 80 kg (176 lb), 6-ft-tall (72 in. or 183 cm) man with the diagnosis of gram-negative pneumonia. His serum creatinine concentration is 2.1 mg/dL (186 μmol/L) and is stable. Compute a conventional gentamicin dosage regimen (infused over 1 hour) that would provide approximate peak and trough concentrations of Cmax,ss = 8 mg/L (8 mcg/mL; 17 μmol/L) and Cmin,ss = 1.5 mg/L (1.5 mcg/mL; 3.1 μmol/L), respectively. The patient is within 30% of his ideal body weight [IBWmale = 50 + 2.3(72 in − 60) = 78 kg] and has stable renal function, so the Cockcroft-Gault CLcr estimation equation can be used: CLcr est = [(140 − 65 y)80 kg]/[72(2.1 mg/dL)] = 40 mL/min (0.67 mL/s). The patient's weight and estimated CLcr are used to compute his V and k, respectively: V = 0.26 L/kg(80 kg) = 20.8 L; k = 0.00293(40 mL/min) + 0.014 = 0.131 h−1 or t1/2 = (0.693/0.131 h−1) = 5.3 h. The dosage interval and dose for the desired serum concentrations would then be calculated: τ = [(ln 8 mg/L − ln 1.5 mg/L)/0.131 h−1] + 1 h = 13.7 h rounded to 12 h; D = (1 h)(0.131 h−1)(20.8 L)(8 mg/L) [1 − e−(0.131h−1(12h))/1 − e−(0.131h−1(1h))] = 140 mg. Thus, the prescribed dose would be gentamicin 140 mg every 12 hours administered as a 1-hour infusion. If a loading dose were deemed necessary, it would be given as the first dose [LD = (20.8 L)(8 mg/L) = 166 mg rounded to 170 mg infused over 1 hour], and the first maintenance dose would be administered 12 hours (e.g., one dosage interval) later. Using the Hull and Sarrubi nomogram for the same patient, the loading dose is 160 mg (gentamicin loading dose for serious gram-negative infection is 2 mg/kg: 2 mg/kg × 80 kg = 160 mg), and the maintenance dose is 115 mg every 12 hours (for a 12-hour dosage interval and CLcr est = 40 mL/min (0.65 mL/s), maintenance dose is 72% of the loading dose: 0.72 × 160 mg = 115 mg).

For extended-interval therapy, Cmax,ss values of 20 to 30 mg/L (20 to 30 mcg/mL; 42 to 63 μmol/L) and Cmin,ss values less than 1 mg/L (1 mcg/mL; 2 μmol/L) generally are accepted as appropriate for gram-negative pneumonia patients. A minimum 24-hour dosage interval is chosen for this dosing technique, and the dosing interval is increased in 12- to 24-hour increments for patients with renal dysfunction.

An example of this initial dosage scheme for the same case is provided to illustrate the use of extended-interval dosing. Mr. JJ is 65 years old, weighs 80 kg (176 lb). His height is 6 ft (72 in. [183 cm]) and his diagnosis is gram-negative pneumonia. His serum creatinine concentration is 2.1 mg/dL (186 μmol/L) and is stable. Compute an extended-interval gentamicin dosage regimen (infused over 1 hour) that would provide approximate peak and trough concentrations of Cmax,ss = 25 mg/L (25 mcg/mL; 52 μmol/L) and Cmin,ss = 0.5 mg/L (0.5 mcg/mL; 1 μmol/L), respectively. The patient is within 30% of his ideal body weight [IBWmale = 50 + 2.3(72 in − 60) = 78 kg] and has stable renal function, so the Cockcroft-Gault CLcr estimation equation can be used: CLcr est = [(140 − 65 y)80 kg]/[72(2.1 mg/dL)] = 40 mL/min (0.67 mL/s). The patient's weight and estimated CLcr are used to compute his V and k, respectively: V = 0.26 L/kg(80 kg) = 20.8 L; k = 0.00293(40 mL/min) + 0.014 = 0.131 h−1 or t1/2 = (0.693/0.131 h−1) = 5.3 h. The dosage interval and dose for the desired serum concentrations would then be calculated: τ = [(ln 25 mg/L − ln 0.5 mg/L)/0.131 h−1] + 1 h = 31 h rounded to 36 h; D = (1 h)(0.131 h−1)(20.8 L)(25 mg/L) [1 − e−(0.131h−1(36h))/1 − e−(0.131h−1(1h))] = 550 mg. Thus, the prescribed dose would be gentamicin 550 mg every 36 hours administered as a 1-hour infusion.

If appropriate aminoglycoside serum concentrations are available, kinetic parameters can be calculated at any point in therapy. When the patient is not at steady state, serum aminoglycoside concentrations are obtained before a dose (Cmin), after a dose administered as an IV infusion of ˜1 hour or as a 30-minute infusion followed by a 30-minute waiting period to allow for drug distribution (Cmax), and at one additional postdose time (C3) approximately one estimated half-life after Cmax. The t1/2 and k values are computed using Cmax and C3: k = (ln Cmax − ln C3)/Δt and t1/2 = 0.693/k, where Δt is the time that expired between the times Cmax and C3 were obtained. If the patient is at steady state, serum aminoglycoside concentrations are obtained before a dose (Cmin,ss) and after a dose administered as an IV infusion of ˜1 hour or as a 30-minute infusion followed by a 30-minute waiting period to allow for drug distribution (Cmax,ss). Because the patient is at steady state, it can be assumed that Cmin,ss is identical for each dosage interval. The t1/2 and k values are computed using Cmax,ss and Cmin,ss: k = (ln Cmax,ss − ln Cmin,ss)/(τ − T) and t1/2 = 0.693/k, where τ is the dosage interval, and T is the dose infusion time or dose infusion time plus waiting time.

Assuming a one-compartment model, the following equation is used to compute VD:35

where D is the dose, and T is the duration of infusion. Once these are known, the dose and dosage interval (τ) can be calculated for any desired maximum Css (Cmax,ss) and minimum Css (Cmin,ss):

The dose and dosage interval should be rounded to provide clinically accepted values (every 8, 12, 18, 24, 36, and 48 hours for dosage interval, nearest 5 to 10 mg for conventional dosing; every 24, 36, and 48 hours for dosage interval, nearest 10 to 25 mg for extended interval dosing). This method also has been used to individualize IV theophylline dosage regimens.40

To provide an example of this technique, the problem given previously for conventional dosing will be extended to include steady-state concentrations. Please note that this method of dosage adjustment using serum concentrations can also be used for extended-interval dosing. Mr. JJ was prescribed gentamicin 140 mg every 12 hours (infused over 1 hour) for the treatment of gram-negative pneumonia. Steady-state trough (Cmin,ss) and peak (Cmax,ss) values were obtained before and after the fourth dose was given (more than three to five estimated half-lives), respectively, and equaled Cmin,ss = 2.8 mg/L (2.8 mcg/mL; 5.9 μmol/L) and Cmax,ss = 8.5 mg/L (8.5 mcg/mL; 18 μmol/L). Clinically, the patient was improving with decreased white blood cell counts and body temperatures and a resolving chest radiograph. However, the serum creatinine value had increased to 2.5 mg/dL (221 μmol/L). Because of this, a new dosage regimen with a similar peak (to maintain high intrapulmonary levels) but lower trough (to decrease the risk of drug-induced nephrotoxicity) concentrations was suggested. The patient's elimination rate constant and half-life can be computed using the following formulas: k = (ln 8.5 mg/L − ln 2.8 mg/L)/(12 h − 1 h) = 0.101 h−1 and t1/2 = 0.693/0.101 h−1 = 6.9 h. The patient's volume of distribution can be calculated using the following equation:

Thus, the patient's volume of distribution was larger and half-life was longer than originally estimated; this led to higher serum concentrations than anticipated. To achieve the desired serum concentrations (Cmin,ss = 1.5 mg/L [1.5 mcg/mL; 3.1 μmol/L] and Cmax,ss = 8 mg/L [8.0 mcg/mL; 17 μmol/L]), the patient's actual kinetic parameters are used to compute a new dose and dosage interval: τ = [(ln 8 mg/L − ln 1.5 mg/L)/0.101 h−1] + 1 h = 17.6 h, rounded to 18 h and

Thus, the new dose would be gentamicin 160 mg every 18 hours and infused over 1 hour; the first dose of the new dosage regimen would be given 18 hours (e.g., the new dosage interval) after the last dose of the old dosage regimen.

Because aminoglycoside antibiotics exhibit concentration-dependent bacterial killing, and the postantibiotic effect is longer with higher concentrations, investigators studied the possibility of giving a higher dose of aminoglycoside using an extended-dosage interval (24 hours or longer, depending on renal function). Generally, these studies have shown comparable microbiologic and clinical cure rates for many infections and about the same rate of nephrotoxicity (˜5−10%) as with conventional dosing. Ototoxicity has not been monitored using audiometry in most of these investigations, but loss of hearing in the conversational range, as well as signs and symptoms of vestibular toxicity, usually has been assessed and found to be similar to that with aminoglycoside therapy dosed conventionally. Based on these data, clinicians are using extended-interval dosing in selected patients. For Pseudomonas aeruginosa infections where the organism has an expected minimum inhibitory concentration (MIC) ≈ 2 mg/L, peak concentrations between 20 and 30 mg/L (20 and 30 mcg/mL) and trough concentrations less than 1 mg/L (1 mcg/mL; 2 μmol/L) for gentamicin or tobramycin have been suggested.41

At the present time, there is no consensus on how to approach concentration monitoring using this mode of administration. Some clinicians obtain steady-state peak and trough concentrations and use the kinetic equations given earlier to adjust the dose and dosage interval in order to attain appropriate target levels. Other clinicians measure only trough concentrations, trusting that the large doses administered to patients achieve adequate peak concentrations.

Also, a nomogram that adjusts extended-interval doses based on a single postdose concentration to achieve these Css goals has been proposed (eFig. 5-12). The dose is 7 mg/kg of gentamicin or tobramycin. The initial dosage interval is set according to the patient's CLcr. The Hartford nomogram includes a method to adjust doses based on serum concentrations. This portion of the nomogram contains average serum concentration time lines for gentamicin or tobramycin in patients with creatinine clearances of 60, 40, and 20 mL/min (1, 0.67, 0.33 mL/s, respectively). A serum concentration is measured 6 to 14 hours after the first dose is given, and this concentration/time point is plotted on the graph (see eFig. 5-12). The modified dosage interval is indicated by which zone the serum concentration/time point falls. Because cystic fibrosis patients have a different volume of distribution (0.35 L/kg) than assumed by this dosing technique, and extended-interval dosing has not been tested adequately in patients with endocarditis, the Hartford nomogram should not be used in these situations.

To illustrate how the Hartford nomogram is used, the same patient example used previously will be repeated for this dosage approach. Mr. JJ weighs 80 kg (176 lb) and has a CLcr est of 40 mL/min (0.67 mL/s). Using the Hartford nomogram, the patient would receive gentamicin 560 mg every 36 hours (7 mg/kg × 80 kg = 560 mg; the initial dosage interval for CLcr est = 40 mL/min [0.67 mL/s] is 36 hours). Ten hours after the first dose was given, the serum gentamicin concentration is 8.2 mg/L (17 μmol/L). According to the graph contained in the nomogram, the dosage interval should be changed to 48 hours. The new dose is 560 mg every 48 hours.

Sidebar: Clinical Controversy…

Some clinicians use conventional dosing or extended-interval dosing exclusively for patients requiring aminoglycosides, whereas others use a mix of both approaches according to the perceived benefit to the patient. Definitive, authoritative recommendations to guide the choice of one method of aminoglycoside dosing over the other are not available.

Sidebar: Clinical Controversy…

“Trough only” measurement of steady-state vancomycin concentrations is a mainstream method to monitor therapy. The exact range for this value is uncertain. Many clinicians recommend 5 to 15 mcg/mL (5 to 15 mg/L; 3.45 to 10 μmol/L). For some sites of infection with specific organisms, such as hospital-acquired pneumonia caused by multidrug-resistant organisms, guidelines suggest vancomycin trough concentrations as high as 15 to 20 mcg/mL (15 to 20 mg/L; 10 to 14 μmol/L) may be necessary. Some clinicians continue to measure both steady-state peak and trough vancomycin concentrations. Optimal outcomes are usually associated with AUC24/MIC ratios greater than 400, where MIC is the minimum inhibitory concentration for the causative organism.

Vancomycin

Vancomycin requires multicompartment models to completely describe its serum-concentration-versus-time curves. However, if peak serum concentrations are obtained after the distribution phase is completed (usually 30 minutes to 1 hour after a 1-hour IV infusion), a one-compartment model can be used for patient dosage calculations. Also, because vancomycin has a relatively long half-life compared with the infusion time, only a small amount of drug is eliminated during infusion, and it is usually unnecessary to use more complex IV infusion equations. Thus, simple IV bolus equations can be used to calculate vancomycin doses for most patients. Although a recent review article42 questioned the clinical usefulness of measuring vancomycin concentrations on a routine basis, other research articles44, 45 have shown potential benefits in obtaining vancomycin concentrations in select patient populations. Some clinicians advocate monitoring only steady-state trough concentrations of vancomycin.46 The decision to conduct vancomycin concentration monitoring should be made on a patient-by-patient basis.

Initial doses of vancomycin can be computed for adult patients using estimated kinetic parameters derived from population pharmacokinetic data. Clearance is estimated using the patient's creatinine clearance in the following equation:43 CL (in mL/min/kg) = 0.695(CLcr in mL/min/kg) + 0.05. The volume of distribution is computed assuming the standard value of 0.7 L/kg: VD = 0.7(Wt), where Wt is the patient's weight. In the case of obese patients, actual or total body weight is used in the calculation of clearance, but ideal body weight is used to compute volume of distribution.47 The elimination rate constant is calculated using clearance and volume of distribution estimates, correcting for possible differences in units for these parameters: k = CL/VD. A nomogram that uses this type of approach for vancomycin therapy is available to determine initial doses rapidly for patients (eTable 5-6).24

Steady-state peak and trough concentrations are chosen for the patient based on the site and severity of the infection, as well as the known or suspected pathogen and avoidance of potential side effects. Optimal outcomes are usually associated with AUC24/MIC ratios greater than 400, where AUC24 is the area under the vancomycin concentration/time curve for 24 hours and MIC is the minimum inhibitory concentration for the bacteria causing the infection. Cmax,ss values of between 20 and 40 mg/L (20 and 40 mcg/mL; 14 and 28 μmol/L) and Cmin,ss values of between 5 and 15 mg/L (5 and 15 mcg/mL; 3 and 10 μmol/L) typically are used for patients with mild to moderate infections or sensitive bacteria with lower MIC values (<1 mcg/mL). For patients with pneumonia or other life-threatening infections due to multidrug-resistant organisms, Cmin,ss as high as 15–20 mg/L (15–20 mcg/mL; 10–14 μmol/L) have been suggested.48 After appropriate steady-state concentrations are chosen, the dosage interval required to attain those concentrations is computed, and τ is rounded to a clinically acceptable value (8, 12, 18, 24, 36, 48, or 72 hours): τ = (ln Cmax,ss − ln Cmin,ss)/k. Finally, the maintenance dose is computed for the patient using a one-compartment-model IV bolus equation at steady state, and the dose is rounded off to the nearest 100 to 250 mg:

If desired, a loading dose can be computed using the following equation:

The following case will illustrate the use of this dosage methodology. Ms. HJ is 65 years old, weighs 68 kg (150 lb), and is 5 ft 4 in. (64 in. [163 cm]) tall. She has developed a surgical wound infection; S. epidermidis is the suspected pathogen. Her serum creatinine concentration is 1.8 mg/dL (159 μmol/L) and stable. Compute a vancomycin dosage regimen that would provide approximate peak (obtained 1 hour after a 1-hour infusion) and trough concentrations of 30 and 7 mg/L (30 and 7 mcg/mL; 21 and 5 μmol/L), respectively. The patient is within 30% of her ideal body weight [IBWfemale = 45 + 2.3(64 in. − 60) = 54 kg] and has stable renal function, so the Cockcroft-Gault creatinine clearance estimation formula can be used: CLcr est = 0.85[(140 − 65 y)68 kg]/[72(1.8 mg/dL)] = 33 mL/min (0.55 mL/s). The patient's weight and CLcr est are used to calculate her estimated CL, VD, and k, respectively: CL = 0.695 (33 mL/min/68 kg) + 0.05 = 0.387 mL/min/kg; VD = 0.7 L/kg (68 kg) = 48 L; and k = [(0.387 mL/min/kg)(68 kg)(60 min/h)]/[(48 L)(1,000 mL/L)] = 0.033 h−1 or t1/2 = 0.693/0.033 h−1 = 21 h. The dosage interval, maintenance dose, and loading dose for the desired serum concentrations can be computed: τ = (ln 30 mg/L − ln 7 mg/L)/0.033 h−1 = 44 h, rounded to 48 h; D = (30 mg/L) (48 L)(1 − e−(0.033h−1)(48h)) = 1,145 mg, rounded to 1,250 mg; LD = (48 L)(30 mg/L) = 1,440 mg, rounded to 1,500 mg. Therefore, the prescribed dose would be vancomycin 1,250 mg every 48 hours administered as a 1-hour infusion. If a loading dose was used, it would be given as the first dose, and the first maintenance dose would be administered 48 hours (one dosage interval) later. Using the Matzke nomogram for the same patient, the loading dose would be 1,750 mg (vancomycin loading dose is 25 mg/kg: 25 mg/kg × 68 kg = 1,700 mg, rounded to 1,750 mg), followed by a maintenance dose of 1,250 mg every 48 hours (for CLcr est = 30 mL/min (0.50 mL/s)), the maintenance dose is 19 mg/kg every 2 days: 19 mg/kg × 68 kg = 1,292 mg, rounded to 1,250 mg).

If appropriate vancomycin serum concentrations are available, kinetic parameters can be computed at any point in therapy. When the patient is not at steady state, serum vancomycin concentrations are obtained before a dose (Cmin), after a dose administered as an IV infusion of 1 hour followed by a 30-minute to 1-hour waiting period to allow for drug distribution (Cmax), and at one additional postdose time (C3) approximately one estimated half-life after Cmax. The t1/2 and k values are computed using Cmax and C3: k = (ln Cmax − ln C3)/Δt and t1/2 = 0.693/k, where Δt is the time that expired between the times Cmax and C3 were obtained. If the patient is at steady state, serum vancomycin concentrations are obtained before a dose (Cmin,ss) and after a dose administered as an IV infusion of approximately 1 hour followed by a 30-minute to 1-hour waiting period to allow for drug distribution (Cmax,ss). The t1/2 and k values are computed using Cmax,ss and Cmin,ss: k = (ln Cmax,ss − ln Cmin,ss)/(τ − Tmax) and t1/2 = 0.693/k, where τ is the dosage interval, and Tmax is the dose infusion time plus waiting time.

Assuming a one-compartment model, the following equation is used to compute VD:

where D is dose. Once these are known, the dose and dosage interval (τ) can be calculated for any desired maximum Css(Cmax,ss) and minimum Css(Cmin,ss):

The dose and dosage interval should be rounded to provide clinically accepted values (every 8, 12, 18, 24, 36, 48, or 72 hours for dosage interval, nearest 100 to 250 mg for dose).

To provide an example for this dosage-calculation method, the preceding problem will be extended to include steady-state concentrations. Ms. HJ was prescribed vancomycin 1,200 mg every 48 hours (infused over 1 hour) for the treatment of a surgical wound infection. Steady-state trough (Cmin,ss) and peak (Cmax,ss) values (Cmax,ss obtained 1 hour after the end of the infusion) were obtained before and after the third dose was given (more than three to five estimated half-lives), respectively, and equaled Cmin,ss = 2.5 mg/L (2.5 mcg/mL; 1.7 μmol/L) and Cmax,ss = 22.4 mg/L (22.4 mcg/mL; 15.5 μmol/L). Clinically, the patient had improved somewhat, but her white blood cell count was still elevated, and the patient was still febrile. Because of this, a modified dosage regimen with a Cmax,ss = 30 mg/L (30 mcg/mL; 21 μmol/L) and Cmin,ss = 7 mg/L (7 mcg/mL; 5 μmol/L) was suggested to maintain an AUC24/MIC ratio greater than 400. The patient's actual elimination rate constant and half-life can be calculated using the following formulas: k = (ln 22.4 mg/L − ln 2.5 mg/L)/(48 h − 2 h) = 0.048 h−1 and t1/2 = 0.693/0.048 h−1 = 14.4 h. The patient's volume of distribution can be calculated using the following equation:

Thus, the patient's volume of distribution was larger and half-life shorter than originally estimated; this led to lower serum concentrations than anticipated. To achieve the desired serum concentrations (Cmax,ss = 30 mg/L [30 mcg/mL; 21 μmol/L] and Cmin,ss = 7 mg/L [7 mcg/mL; 5 μmol/L]), the patient's actual kinetic parameters are used to calculate a new dose and dosage interval:

The new dose would be vancomycin 1,500 mg every 36 hours (infused over 1 hour); the first dose of the new dosage regimen would be given 36 hours (the new dosage interval) after the last dose of the old dosage regimen.

For routine monitoring, many clinicians measure only steady-state vancomycin trough concentrations in patients. The justification for this approach is that because vancomycin exhibits time-dependent bacterial killing, the minimum concentration is the most important with regard to therapeutic outcome. Vancomycin pharmacokinetics also support this approach because the volume of distribution is relatively stable and is not changed by many disease states or conditions. Because of this important point, it is difficult to attain peak steady-state concentrations in the toxic range when the steady-state vancomycin trough is in the therapeutic range if typical doses are used (15 mg/kg or ≈ 1,000 mg for average-weight individuals). Also, toxic peak concentrations (generally greater than 80–100 mg/L [80 to 100 mcg/mL; 55 to 69 μmol/L]) are quite a bit higher than therapeutic peak concentrations, which adds a safety margin between effective concentrations and those yielding adverse drug effects.

Coupled with trough-only vancomycin concentration monitoring is a widening of the therapeutic steady-state trough concentration range from 5 to 15 mg/L (5 to 15 mcg/mL; 3 to 10 μmol/L). The justification for increasing the top of the range from 10 to 15 mg/L (10 to 15 mcg/mL; 7 to 10 μmol/L) comes from limited retrospective45 and prospective45 studies. Trough concentrations in the range of 15 to 20 mg/L (15 to 20 mcg/mL; 10 to 14 μmol/L) should be reserved for specific clinical situations, such as hospital-acquired pneumonia or other severe infections caused by multidrug-resistant organisms.48 As previously mentioned, optimal outcomes are usually associated with AUC24/MIC ratios greater than 400, where MIC is the minimum inhibitory concentration for the causative organism.

When trough-only monitoring of vancomycin concentrations is chosen by a clinician, a simple variant of linear pharmacokinetics can be used to adjust the dose (D) and dosage interval (τ): (Dnew/τnew) = (Dold/τold)(Css,new/Css,old), where new and old indicate the new target trough concentration and the old measured trough concentration, respectively. In practice, the dose (typically 1,000–1,500 mg) is often held constant, and only the dosage interval is changed. This equation is an approximation of the actual new steady-state trough concentration that will be attained in the patient because, mathematically, Css,new is an exponential function of τ.

An example of this approach is given in the following case. Mr. MK is 72 years old, weighs 72 kg (158 lb), and measures 5 ft 9 in. (69 in. [175 cm]). He was prescribed vancomycin 1,000 mg every 12 hours (infused over 1 hour) for the treatment of an S. epidermidis central venous catheter infection. A steady-state trough (Cmin,ss) value was obtained before the fifth dose was given (more than three to five estimated half-lives), and Cmin,ss = 19 mg/L (19 mcg/mL; 13 μmol/L). Clinically, the patient was improving, but the trough concentration was judged to be too high. Because of this, a modified dosage regimen with a Cmin,ss = 10 mg/L (10 mcg/mL; 7 μmol/L) was suggested to maintain an AUC24/MIC ratio greater than 400. (Dnew/τnew) = (1,000 mg/12 h)(10 mg/L/19 mg/L) = 44 mg/h. Because the patient is near his ideal weight, the same dose of 1,000 mg can be used (Dnew), and the new dosage interval (τnew) can be computed: τ = 1,000 mg/44 mg/h = 23 h, rounded to 24 h. The new prescribed dose for the patient would be 1,000 mg every 24 hours.

Digoxin

Digoxin pharmacokinetics are best described by a two-compartment model. However, because digoxin has a long half-life compared with its dosage interval and a very long distribution phase, simple pharmacokinetic equations can be used to individualize dosing when postdistribution serum concentrations are used. Digoxin can be given as an IV injection and orally as elixir (F = 0.8) or tablets (F = 0.7). When given orally, the appropriate bioavailability fraction must be used to compute the correct dose. Initial doses of digoxin can be computed using population pharmacokinetic data obtained from published studies. Digoxin clearance is estimated using the patient's CLcr in the following formula:23 CL (in milliliters per minute) = 1.303(CLcr in milliliters per minute) + CLm, where CLm is metabolic clearance and equals 40 mL/min for patients with no or mild heart failure or 20 mL/min for patients with moderate to severe heart failure. The volume of distribution decreases with declining renal function and is estimated using the following equation:23VD (in liters) = 226 + [298(CLcr in milliliters per minute)]/(29.1 + CLcr in milliliters per minute). The elimination rate constant can be computed by taking the product of CL and VD: k = CL/VD. For obese individuals, digoxin dosing should be based on ideal body weight.49

Appropriate Css values are chosen for the patient based on the disease state being treated, the goal of therapy, and avoidance of adverse effects. The inotropic effects of digoxin occur at lower concentrations than do the chronotropic effects. Therefore, initial serum concentrations of digoxin for the treatment of heart failure generally are 1 ng/mL (1 mcg/L; 1.3 nmol/L) or less and for the treatment of atrial fibrillation 1 to 1.5 ng/mL (1 to 1.5 mcg/L; 1.3 to 1.9 nmol/L). Once the appropriate Css is selected, a dose is computed for the patient: D/τ = (CssCL)/F.

An example of this initial dosage scheme is provided in the following case. Mr. PO is 72 years old, weighs 83 kg (183 lb), and measures 5 ft 11 in. (71 in. [180 cm]). He was admitted to the hospital for the treatment of community-acquired pneumonia. His past medical history is positive for moderate heart failure. While in the hospital, Mr. PO develops atrial fibrillation, and the decision is made to treat him with digoxin to provide ventricular rate control. His serum creatinine concentration is 2.5 mg/dL (221 μmol/L) and stable. Calculate an IV loading dose and oral maintenance dose that will achieve a Css of 1.5 ng/mL (1.5 mcg/L; 1.9 nmol/L). The Cockcroft-Gault equation can be used to estimate the patient's CLcr because his serum creatinine concentration is stable, and he is within 30% of his ideal body weight [IBWmale = 50 + 2.3(71 in − 60) = 75 kg]: CLcr = [(140 − 72 y)83 kg]/[72(2.5 mg/dL)] = 31 mL/min (0.52 mL/s). Using the estimated CLcr, both CL and VD can be computed:

The maintenance dose will be given as digoxin tablets, so F = 0.7 in the dosing equation: D/τ = [(1.5 mcg/L)(60 mL/min)(60 min/h)(24 h/day)]/[0.7(1,000 mL/L)] = 185 mcg/day, rounded to 187.5 mcg/day (given as 1 1/2 of 125 mcg tablets). The loading dose will be given IV as a digoxin injection: LD = (1.5 mcg/L)(380 L) = 570 mcg, rounded to 500 mcg. The loading dose would be given 50% now (250 mcg), 25% (125 mcg) in 4 to 6 hours after monitoring the patient's heart rate and blood pressure and assessing the patient for digoxin adverse effects, and the final 25% (125 mcg) 4 to 6 hours later after monitoring the same clinical parameters. The first maintenance dose would be given one dosage interval (in this case, 24 hours) after the first part of the loading dose was given.

Adjustment of digoxin doses using steady-state concentrations is accomplished using linear pharmacokinetics and dosage ratios: Dnew = Dold(Css,new/Css,old). For example, Mr. PO's atrial fibrillation responded to digoxin therapy, and he was discharged after resolution of his pneumonia. A month later, he was followed up in the clinic with moderate nausea, possibly a result of digoxin toxicity. His heart rate was 51 beats per minute, and his serum creatinine was unchanged. A steady-state digoxin concentration was determined and reported by the clinical laboratory as 2.2 mcg/L (2.8 nmol/L). Compute a new dose for the patient to achieve a Css of 1.5 mcg/L (1.9 nmol/L). The digoxin Css and old dose would be used to calculate a new dose using the linear pharmacokinetic equation: Dnew = 187.5 mcg/day[(1.5 mcg/L)/(2.2 mcg/L)] = 128 mcg/day, rounded to 125 mcg/day.

Theophylline

Theophylline disposition is described most accurately by nonlinear kinetics.50, 51 However, at the usual doses, theophylline acts as if it obeys linear kinetics in most patients. Initial theophylline doses are computed by taking a detailed medical history of the patient and noting disease states and conditions that are known to change theophylline disposition. Age, smoking of tobacco-containing products, heart failure, and liver disease are among the important factors that alter theophylline kinetic parameters and dosage requirements. Once the patient has been assessed, average theophylline kinetic parameters obtained from the literature for patients similar to the one being currently treated are used to compute either oral or IV doses. Dosage guidelines that take into account most common disease states and conditions that change theophylline kinetic parameters are available (see eTable 5-4).52 Once theophylline is administered, the patient is monitored for the therapeutic effect and potential adverse effects. Theophylline concentrations then are used to individualize the theophylline dose that the patient receives. An example of this approach was given previously for a patient in the section on drug dosing in patients with liver disease (see Selection of Initial Drug Doses above).

Continuous IV infusions of theophylline (or its salt, aminophylline) can be individualized rapidly by determining the patient's CL before steady state occurs.53 Assuming that the patient receives theophylline only by continuous IV infusion (previous doses of sustained-release oral theophylline are completely absorbed), two serum theophylline concentration determinations are done 4 hours or more apart. The infusion rate (k0) cannot be changed between the times the samples are drawn. With one-compartment model equations, the first (C1) and second (C2) theophylline concentrations are used to calculate theophylline CL:

VD is assumed to be 0.5 L/kg, and t1 and t2 are the times at which C1 and C2, respectively, are obtained. Once CL is known, k0 can be computed easily for any desired Css (Css = k0/CL). This method probably can be applied to other drugs that are administered as continuous IV infusions, such as IV antiarrhythmics, when rapid individualization of drug dosage is desirable.

An example of this approach can be obtained by continuing the theophylline patient case from the section on drug dosing in liver disease (see Selection of Initial Drug Doses above). In this example, a 55-year-old, 70 kg (154 lb) man with liver cirrhosis was prescribed a loading dose of theophylline 350 mg IV over 20 to 30 minutes, followed by a maintenance dose of 15 mg/h of theophylline as a continuous infusion. The infusion began at 9 AM, blood samples were obtained at 10 AM and 4 PM, and the clinical laboratory reported the theophylline serum concentrations as 10.9 and 12.3 mg/L (10.9 and 12.3 mcg/mL; 60.5 and 68.3 μmol/L), respectively. The patient's theophylline clearance and revised continuous infusion to maintain a Css of 15 mg/L (15 mcg/mL; 83.3 μmol/L) can be computed as follows (patient's VD estimated at 0.5 L/kg):

If theophylline is to be given as the aminophylline salt form, the doses would need to be changed to reflect the fact that aminophylline contains only 85% theophylline (k0 = 9 mg/h theophylline/ 0.85 = 11 mg/h aminophylline).

If continuous IV infusions or oral dosage regimens are given long enough for steady state to occur (three to five estimated half-lives based on previous studies conducted in similar patients), linear pharmacokinetics can be used to adjust doses for either route of administration: Dnew = Dold(Css,new/Css,old). For example, a patient receiving 200 mg of sustained-release oral theophylline every 12 hours with a theophylline steady-state serum concentration of 9.5 mcg/mL (9.5 mg/L; 52.7 μmol/L) can have the dose required to achieve a new Css equal to 15 mcg/mL (15 mg/L; 83.3 μmol/L) computed by applying linear pharmacokinetics: Dnew = 200 mg[(15 mcg/mL)/(9.5 mcg/mL)] = 316 mg, rounded to 300 mg. Thus the new theophylline dose would be 300 mg every 12 hours.

Phenytoin

Phenytoin doses are very difficult to individualize because the drug follows Michaelis-Menten kinetics, and there is a large amount of interpatient variability in Vmax and Km. Initial maintenance doses of phenytoin in adults usually range between 4 and 7 mg/kg daily, yielding starting doses of 300 to 400 mg daily in most individuals. If needed, loading doses of phenytoin or fosphenytoin (a prodrug of phenytoin used IV) can be administered in adults at a dose of 15 mg/kg, which is approximately 1,000 mg in many individuals. Loading doses of phenytoin can be given orally but need to be administered in divided doses separated by several hours in order to avoid decreased bioavailability and gastrointestinal intolerance (for total loading dose of 1,000 mg: 400 mg, 300 mg, then 300 mg with each dose separated by 4 to 6 h). Because phenytoin is metabolized hepatically, decreased doses may be needed in patients with liver disease. Because phenytoin follows dose-dependent pharmacokinetics, the half-life of phenytoin increases for a patient as the maintenance dose increases. Therefore, the time to steady-state phenytoin concentrations increases with dose. On average, at a phenytoin dose of 300 mg daily, it takes approximately 5 to 7 days to achieve steady state; at a dose of 400 mg daily, it takes approximately 10 to 14 days to achieve steady state; and at a dose of 500 mg daily, it takes approximately 21 to 28 days to achieve steady state. It should be noted that the injectable and capsule dosage forms of phenytoin are phenytoin sodium, and the labeled dosage amounts contain 92% of active phenytoin (300 mg phenytoin sodium capsules contain 276 mg [300 mg × 0.92 = 276 mg] of active phenytoin). Unbound phenytoin concentrations are useful in patients with hypoalbuminemia (e.g., liver disease, nephrotic syndrome, pregnancy, cystic fibrosis, burns, trauma, and malnourishment, as well as in the elderly), in patients in whom displacement with endogenous compounds is possible (e.g., hyperbilirubinemia, liver disease, or end-stage renal disease), and in patients receiving other drugs that may displace phenytoin from plasma protein-binding sights (e.g., valproic acid, aspirin therapy >2 g daily, warfarin, and nonsteroidal antiinflammatory drugs with high albumin binding).54

After steady state has occurred, phenytoin serum concentrations can be obtained as an aid to dosage adjustment. A simple, easy way to approximate new serum concentrations after a dosage adjustment with phenytoin is to temporarily assume linear pharmacokinetics and then add 15% to 33% for a dosage increase or subtract 15% to 33% for a dosage decrease to account for Michaelis-Menten kinetics. To avoid large disproportionate changes in phenytoin concentrations when using this empirical method, dosage adjustments should be limited to 50 to 100 mg daily. This technique is intended only to provide a rough approximation of the resulting phenytoin Css after an appropriate dosage adjustment has been made.

For example, Ms. PP is a 35-year-old, 65 kg (143 lb) patient with grand mal seizures who is receiving phenytoin capsules 300 mg orally at bedtime. A Css of 9.2 mcg/mL (9.2 mg/L; 37 μmol/L) is measured. It is observed that her seizure frequency decreased by only ˜15%, and that she has had no adverse effects as a consequence of phenytoin treatment. Because of this, her phenytoin dose is increased to 400 mg orally at bedtime. The expected phenytoin Css would be estimated using linear pharmacokinetics [Cnew = (Dnew/Dold)Cold = (400 mg/300 mg)/(9.2 mcg/mL) = 12.3 mcg/mL] and then increased by 15% to 33% to account for nonlinear kinetics [Cnew = 1.15(12.3 mcg/mL) = 14.1 mcg/mL or Cnew = 1.33 (12.3 mcg/mL) = 16.4 mcg/mL]. Thus, the patient would be expected to have a steady-state phenytoin concentration of approximately 14 to 16 mcg/mL (14 to 16 mg/L; 56 to 63 μmol/L) as a consequence of the dosage increase. An alternative approach would be to use a graphic Bayesian method that allows an estimate of Vmax and Km from one steady-state phenytoin concentration and the prediction of new steady-state concentrations when doses are changed.55

Other methods used to individualize phenytoin doses involve rearrangements of the Michaelis-Menten equation [DR = VmaxCss/(Km + Css), in which DR is the dosage rate at steady state] so that two or more doses and Css values can be used to obtain graphic solutions for Vmax and Km. One rearrangement56 is DR = −Km(DR/Css) + Vmax. When DR is plotted on the y axis, and DR/Css is plotted on the x axis of Cartesian graph paper, a straight line with a y intercept of Vmax and slope equal to −Km is found (eFig. 5-13). To use this method, patients are prescribed an initial phenytoin dose, and Css is obtained. The phenytoin dose is then changed, and a second Css from the new dose is obtained. Each dose is divided by its respective Css to derive DR/Css values. The DR/Css and Css values are plotted on the graph to calculate Vmax (y intercept) and Km (minus slope). The steady-state Michaelis-Menten equation can be used to compute Css.

Cyclosporine

Because of the large amount of variability in cyclosporine pharmacokinetics, even when concurrent disease states and conditions are identified, many clinicians believe that the use of standardized initial cyclosporine doses for various situations is warranted. Indeed, most transplant centers use doses that are determined employing a locally derived cyclosporine dosage protocol. The original computations of these doses were based on the pharmacokinetic dosing methods described in preceding sections and subsequently modified based on clinical experience. In general, the expected cyclosporine Css used to compute these doses depends on the type of transplanted tissue and the posttransplantation time line. Generally speaking, initial oral doses of 8 to 18 mg/kg daily or IV doses of 3 to 6 mg/kg daily (one-third the oral dose to account for ˜30% oral bioavailability) are used and vary greatly from institution to institution. For obese individuals (>30% over ideal body weight), ideal body weight should be used to compute initial doses.

It is likely that doses computed using patient population characteristics will not always produce cyclosporine concentrations that are expected or desirable. Additionally, there is a very high amount of interday variation in cyclosporine concentrations. Because of pharmacokinetic variability, the narrow therapeutic index of cyclosporine, and the severity of cyclosporine adverse side effects, measurement of cyclosporine concentrations is mandatory for patients to ensure that therapeutic, nontoxic levels are present. When cyclosporine concentrations are measured in patients, and a dosage change is necessary, clinicians should seek to use the simplest, most straightforward method available to determine a dose that will provide safe and effective treatment. In most cases, a simple dosage ratio can be used to change cyclosporine doses using steady-state concentrations and assuming that the drug follows linear pharmacokinetics:

The Css can be either a steady-state trough concentration or a Css measured 2 hours (±15 min) after a dose (C2). When C2 levels are used, recommended concentrations vary according to transplant type and posttransplant time (see eTable 5-7).57–59

For example, LK is a 50-year-old, 75 kg (165 lb), 5 ft 11 in. (71 in. [180 cm]) male renal transplant recipient who is receiving oral cyclosporine 400 mg every 12 hours. The current steady-state blood cyclosporine concentration is 375 ng/mL (375 mcg/L; 312 nmol/L). To compute a cyclosporine dose that will provide a Css of 200 ng/mL (200 mcg/L; 166 nmol/L), linear pharmacokinetic equations can be used. The new dose to attain the desired concentration should be proportional to the old dose that produced the measured concentration (total daily dose = 400 mg/dose × 2 doses/d = 800 mg/d):

The new suggested dose would be 400 mg/day or 200 mg every 12 hours of cyclosporine capsules to be started at the next scheduled dosing time.

Pharmacodynamics is the study of the relationship between the concentration of a drug and the response obtained in a patient. Originally, investigators examined the dose–response relationship of drugs in humans but found that the same dose of a drug usually resulted in different concentrations in individuals because of pharmacokinetic differences in clearance and volume of distribution. Examples of quantifiable pharmacodynamic measurements include changes in blood pressure during antihypertensive drug therapy, decreases in heart rate during β-blocker treatment, and alterations in prothrombin time or international normalized ratio during warfarin therapy.

For drugs that exhibit a direct and reversible effect, the following diagram describes what occurs at the level of the drug receptor:

According to this scheme, there is a drug receptor located within the target organ or tissue. When a drug molecule “finds” the receptor, it forms a complex that causes the pharmacologic response to occur. The drug and receptor are in dynamic equilibrium with the drug–receptor complex.

The Emax and Sigmoid Emax Models

The mathematical model that comes from the classic drug receptor theory shown previously is known as the Emax model:

where E is the pharmacologic effect elicited by the drug, Emax is the maximum effect the drug can cause, EC50 is the concentration causing one-half the maximum drug effect (Emax/2), and C is the concentration of drug at the receptor site. EC50 can be used as a measure of drug potency (a lower EC50, indicating a more potent drug), whereas Emax reflects the intrinsic efficacy of the drug (a higher Emax, indicating greater efficacy). If pharmacologic effect is plotted against concentration in the Emax equation, a hyperbola results with an asymptote equal to Emax (eFig. 5-14). At a concentration of zero, no measurable effect is present.

When dealing with human studies in which a drug is administered to a patient, and pharmacologic effect is measured, it is very difficult to determine the concentration of the drug at the receptor site. Because of this, serum concentrations (total or unbound) usually are used as the concentration parameter in the Emax equation. Therefore, the values of Emax and EC50 are much different than if the drug were added to an isolated tissue contained in a laboratory beaker.

The result is that a much more empirical approach is used to describe the relationship between concentration and effect in clinical pharmacology studies. After a pharmacodynamic experiment has been conducted, concentration–effect plots are generated. The shape of the concentration–effect curve is used to determine which pharmacodynamic model will be used to describe the data. Because of this, the pharmacodynamic models used in a clinical pharmacology study are deterministic in the same way that the shape of the serum-concentration-versus-time curve determines which pharmacokinetic model is used in clinical pharmacokinetic studies.

Sometimes a hyperbolic function does not describe the concentration–effect relationship at lower concentrations adequately.

When this is the case, the sigmoid Emax equation may be superior to the Emax model:

where n is an exponent that changes the shape of the concentration–effect curve. When n > 1, the concentration–effect curve is S- or sigmoid-shaped at lower serum concentrations. When n < 1, the concentration–effect curve has a steeper slope at lower concentrations (eFig. 5-15).

With both the Emax and sigmoid Emax models, the largest changes in drug effect occur at the lower end of the concentration scale. Small changes in low serum concentrations cause large changes in effect. As serum concentrations become larger, further increases in serum concentration result in smaller changes in effect. Using the Emax model as an example and setting Emax = 100 units and EC50 = 20 mg/L, doubling the serum concentration from 5 to 10 mg/L increases the effect from 20 to 33 units (a 67% increase), whereas doubling the serum concentration from 40 to 80 mg/L only increases the effect from 67 to 80 units (a 19% increase). This is an important concept for clinicians to remember when doses are being titrated in patients.

Linear Models

When serum concentrations obtained during a pharmacodynamic experiment are between 20% and 80% of Emax, the concentration–effect curve may appear to be linear (eFig. 5-16). This occurs often because lower drug concentrations may not be detectable with the analytic technique used to assay serum samples, and higher drug concentrations may be avoided to prevent toxic side effects. The equation used is that of a simple line: E = S × C + I, where E is the drug effect, C is the drug concentration, S is the slope of the line, and I is the y intercept. In this situation, the value of S can be used as a measure of drug potency (the larger the value of S, the more potent the drug). The linear model can be derived from the Emax model. When EC50 is much greater than C, E = (Emax/EC50)C = S × C, where S = Emax/EC50.

The linear model allows a nonzero value for effect when the concentration equals zero. This may be a baseline value for the effect that is present without the drug, the result of measurement error when determining effect, or model misspecification. Also, this model does not allow the prediction of a maximum response.

Some investigators have used a log-linear model in pharmacodynamic experiments: E = S × (log C) + I, where the symbols have the same meaning as in the linear model. The advantages of this model are that the concentration scale is compressed on concentration–effect plots for experiments where wide concentration ranges were used, and the concentration values are transformed so that linear regression can be used to compute model parameters. The disadvantages are that the model cannot predict a maximum effect or an effect when the concentration equals zero. With the increased availability of nonlinear regression programs that can compute the parameters of nonlinear functions such as the Emax model easily, use of the log-linear model has been discouraged.60

Baseline Effects

At times, the effect measured during a pharmacodynamic study has a value before the drug is administered to the patient. In these cases, the drug changes the patient's baseline value. Examples of these types of measurements are heart rate and blood pressure. In addition, a given drug may increase or decrease the baseline value. Two basic techniques are used to incorporate baseline values into pharmacodynamic data. One way incorporates the baseline value into the pharmacodynamic model; the other transforms the effect data to take baseline values into account.

Incorporation of the baseline value into the pharmacodynamic model involves the addition of a new term to the previous equations. E0 is the symbol used to denote the baseline value of the effect that will be measured. The form that these equations takes depends on whether the drug increases or decreases the pharmacodynamic effect. When the drug increases the baseline value, E0 is added to the equations:

When E0 is not known with any better certainty than any other effect measurement, it should be estimated as a model parameter similar to the way that one would estimate the values of Emax, EC50, S, or n.61, 62 If the baseline effect is well known and has only a small amount of measurement error, it can be subtracted from the effect determined in the patient during the experiment and not estimated as a model parameter. This approach can lead to better estimates of the remaining model parameters.62 Using the linear model as an example, the equation used would be E − E0 = S × C.

If the drug decreases the baseline value, the drug effect is subtracted from E0 in the pharmacodynamic models:

where Emax represents the maximum reduction in effect caused by the drug, and IC50 is the concentration that produces a 50% inhibition of Emax. These forms of the equations have been called the inhibitory Emax and inhibitory sigmoidal Emaxequations, respectively. In this arrangement of the pharmacodynamic model, E0 is a model parameter and can be estimated. If the baseline effect is well known and has little measurement error, the effect in the presence of the drug can be subtracted from the baseline effect and not estimated as a model parameter. Using the inhibitory Emax model as an example, the formula would be E0 − E = (Emax × C)/(IC50 + C).

When using the inhibitory Emax model, a special situation occurs if the baseline effect can be obliterated completely by the drug (e.g., decreased premature ventricular contractions during antiarrhythmic therapy). In this situation, Emax = E0, and the equation simplifies to a rearrangement known as the fractional Emaxequation:

This form of the model relates drug concentration to the fraction of the maximum effect.

An alternative approach to the pharmacodynamic modeling of drugs that alter baseline effects is to transform the effect data so that they represent a percentage increase or decrease from the baseline value.62 For drugs that increase the effect, the following transformation equation would be used: percent effectt = [(treatmentt − baseline)/baseline] × 100. For drugs that decrease the effect, the following formula would be applied to the data: percent inhibitiont = [(baseline − treatmentt)/baseline] × 100. The subscript indicates the treatment, effect, or inhibition that occurred at time t during the experiment. If the study included a placebo control phase, baseline measurements made at the same time as treatment measurements (i.e., heart rate determined 2 hours after placebo and 2 hours after drug treatment) could be used in the appropriate transformation equation.62 The appropriate model (excluding E0) then would be used.

Hysteresis

Concentration–effect curves do not always follow the same pattern when serum concentrations increase as they do when serum concentrations decrease. In this situation, the concentration–effect curves form a loop that is known as hysteresis. With some drugs, the effect is greater when serum concentrations are increasing, whereas with other drugs, the effect is greater while serum concentrations are decreasing (eFig. 5-17). When individual concentration–effect pairs are joined in time sequence, this results in clockwise and counterclockwise hysteresis loops.

Clockwise hysteresis loops usually are caused by the development of tolerance to the drug. In this situation, the longer the patient is exposed to the drug, the smaller is the pharmacologic effect for a given concentration. Therefore, after an extravascular or short-term infusion dose of the drug, the effect is smaller when serum concentrations are decreasing compared with the time when serum concentrations are increasing during the infusion or absorption phase.

Accumulation of a drug metabolite that acts as an antagonist also can cause clockwise hysteresis.

Counterclockwise hysteresis loops can be caused by the accumulation of an active metabolite, sensitization to the drug, or delay in time in equilibration between serum concentration and concentration of drug at the site of action. Combined pharmacokinetic/pharmacodynamic models have been devised that allow equilibration lag times to be taken into account.

The availability of inexpensive, rapidly achievable serum drug concentrations has changed the way clinicians monitor drug therapy in patients. The therapeutic range for many drugs is known, and it is likely that more drugs will be monitored using serum concentrations in the future. Clinicians need to remember that the therapeutic range is merely an average guideline and to take into account interindividual pharmacodynamic variability when treating patients. Individual patients may respond to smaller concentrations or require concentrations that are much greater to obtain a therapeutic effect. Conversely, patients may show toxic effects at concentrations within or below the therapeutic range. Serum concentrations should never replace clinical judgment.

Three kinetic constants determine the dosage requirements of patients. Clearance determines the maintenance dose (MD = CLCss), volume of distribution determines the loading dose (LD = VDCss), and half-life determines the time to steady state and the dosage interval. Several methods are available to compute these parameters.

Methods available to individualize drug therapy range from clinical pharmacokinetic techniques using simple mathematical relationships that hold for all drugs that obey linear pharmacokinetics to very complex computer programs that are specific to one drug.