Chapter 19. Relationship between Pharmacokinetics and Pharmacodynamics

In the previous chapters, pharmacokinetics was used to develop dosing regimens for achieving therapeutic drug concentrations for optimal safety and efficacy. The interaction of a drug molecule with a receptor causes the initiation of a sequence of molecular events, resulting in a pharmacodynamic or pharmacologic response. The term pharmacodynamics refers to the relationship between drug concentrations at the site of action (receptor) and pharmacologic response. Pharmacodynamics includes the biochemical and physiologic effects that result from the interaction of the drug with the receptor. Early pharmacologic research demonstrates that the pharmacodynamic response produced by the drug depends on the chemical structure of the drug molecule and the affinity of the drug at the receptor site. The drug affinity for the receptor site and the resultant pharmacodynamic response is referred to as the intrinsic activity of the drug. Drug receptors interact only with drugs of specific chemical structure, and the receptors are classified according to the type of pharmacodynamic response induced. Drugs may be considered a full agonist, partial agonist, or antagonist, depending upon the type of drug interaction with the receptor and the resulting pharmacodynamic response (see discussion below).

Drug Receptor and Occupancy Concept

Drugs may react with receptors to form covalent or noncovalent bonds. Drugs that form covalent bonds with the receptor produce a nonreversible pharmacodynamic response. Most pharmacologic responses are due to weak, noncovalent bonds (eg, hydrogen bonding, ionic electrostatic bonds, van der Waals forces) between the drug and the receptor. These interactions between drug and receptor are assumed to be reversible and to conform to the law of mass action. One or more drug molecules may interact simultaneously with the receptor to produce a pharmacologic response. Typically, a single drug molecule interacts with a receptor with a single binding site to produce a pharmacologic response, as illustrated below.

where the brackets [ ] denote molar concentrations.

This scheme illustrates the occupation theory for the interaction of a drug molecule with a receptor molecule. More recent schemes consider a drug that binds to macromolecules as a ligand. Thus, the reversible interaction of a ligand (drug) with a receptor may be written as (Neubig et al, 2003).

where L is generally referred to as ligand concentration (since many drugs are small molecules) and LR is analogous to the [drug–receptor complex]. LR* is the activated form which results in the effect.

The last step is written to accommodate different modes of how LR leads to a drug effect. For example, the interaction of a subsequent ligand with the receptor may involve a conformation change of the receptor or simply lead to an additional effect. In this chapter, effect and response are used interchangeably.

This model makes the following assumptions:

1. The drug molecule combines with the receptor molecule as a bimolecular association, and the resulting drug–receptor complex disassociates as a unimolecular entity.

2. The binding of drug with the receptor is fully reversible.

3. The basic model assumes a single type of receptor binding site, with one binding site per receptor molecule. It is also assumed that a receptor with multiple sites may be modeled after this (Taylor and Insel, 1990).

4. The occupancy of the drug molecule at one receptor site does not change the affinity of more drug molecules to complex at additional receptor sites.

Each receptor has equal affinity for the drug molecule. The model is not suitable for drugs with allosteric binding to receptors, in which the binding of one drug molecule to the receptor affects the binding of subsequent drug molecules, as in the case of oxygen molecules binding to iron in hemoglobin. As more receptors are occupied by drug molecules, a greater pharmacodynamic response is obtained until a maximum response is reached.

The receptor occupancy concept was extended to show how drugs elicit a pharmacologic response as an agonist, or block the pharmacologic response as an antagonist through drug– receptor interactions. Basically, three types of related responses may occur at the receptor: (1) a drug molecule that interacts with the receptor and elicits a maximal pharmacologic response is referred to as an agonist; (2) a drug that elicits a partial (below maximal) response is termed a partial agonist; and (3) an agent that elicits no response from the receptor, but inhibits the receptor interaction of a second agent, is termed an antagonist. An antagonist may prevent the action of an agonist by competitive (reversible) or noncompetitive (irreversible) inhibition. A few drugs (eg, pentazocine) have mixedagonist-antagonist activity in that the initial drug binding to a receptor produces a pharmacologic effect followed by blocking the receptor from eliciting additional pharmacologic activity.

Spare, unoccupied receptors are assumed to be present at the site of action, because a maximal pharmacologic response may be obtained when only a small fraction of the receptors are occupied by drug molecules. Equimolar concentrations of different drug molecules that normally bind to the same receptor may give different degrees of pharmacologic response. The term intrinsic activity is used to distinguish the relative extent of pharmacologic response between different drug molecules that bind to the same receptor. The potency of a drug is the concentration of drug needed to obtain a specific pharmacologic effect, such as the EC50 (see Emax model, below).

The receptor occupation theory, however, was not consistent with all kinetic observations. An alternative theory, known as the rate theory, essentially states that the pharmacologic response is not dependent on drug–receptor complex concentration but rather depends on the rate of association of the drug and the receptor. Each time a drug molecule combines with a receptor, a response is produced, similar to a ball bouncing back and forth, to and from the receptor site. The rate theory predicts that an agonist will associate rapidly to form a receptor complex, which dissociates rapidly to produce a response. An antagonist associates rapidly to form a receptor–drug complex and dissociates slowly to maintain the antagonist response.

Both theories are consistent with the observed saturation (sigmoidal) drug–dose response relationships, but neither theory is sufficiently advanced to give a detailed description of the “lock-and-key” or the more recent “induced-fit” type of drug interactions with enzymatic receptors. Newer theories of drug action are based on in vitro studies on isolated tissue receptors and on observation of the conformational and binding changes with different drug substrates. These in vitro studies show that other types of interactions between the drug molecule and the receptor are possible. However, the results from the in vitro studies are difficult to extrapolate to in vivo conditions. The pharmacologic response in drug therapy is often a product of physiologic adaptation to a drug response. Many drugs trigger the pharmacologic response through a cascade of enzymatic events highly regulated by the body.

Unlike pharmacokinetic modeling, pharmacodynamic modeling can be more complex because the clinical measure (change in blood pressure or clotting time) is often a surrogate for the drug's actual pharmacologic action. For example, after the drug is systemically absorbed, it is then transported to site of action where the pharmacologic receptor resides. Drug-receptor binding may then cause a secondary response, such as signal transduction, which then produces the desired effect. Clinical measurement of drug response may only occur after many such biologic events, such as transport or signal transduction (an indirect effect), so pharmacodynamic modeling must account for biologic processes involved in eliciting drug-induced responses.

The complexity of the molecular events triggering a pharmacologic response is less difficult to describe using a pharmacokinetic approach. Pharmacokinetic models allow very complex processes to be simplified. The process of pharmacokinetic modeling continues until a model is found that describes the real process quantitatively. The understanding of drug response is greatly enhanced when pharmacokinetic modeling techniques are combined with clinical pharmacology, resulting in the development of pharmacokinetic–pharmacodynamic models. Pharmacokinetic–pharmacodynamic models use data derived from the plasma drug concentration-versus-time profile and from the time course of the pharmacologic effect to predict the pharmacodynamics of the drug. Pharmacokinetic–pharmacodynamic models have been reported for antipsychotic medications, anticoagulants, neuromuscular blockers, antihypertensives, anesthetics, and many antiarrhythmic drugs (the pharmacologic responses of these drugs are well studied because of easy monitoring).

Drug Receptors

The best-characterized drug receptors are regulatory proteins, which mediate the actions of endogenous chemical signals such as neurotransmitters, autacoids, and hormones. Other protein receptors are endogenous enzymes, which may be inhibited or activated by binding to a drug. For example, the enzyme dihydrofolate reductase is the receptor for the antineoplastic drug methotrexate. The transport protein could also be a class of receptor (eg, Na+, K+ ATPase, the membrane receptor for cardioactive digitalis glycosides).

Various types of drug receptors and pharmacologic responses are described by Katzung et al (2009, Goodman & Gilman, 2006) and quantitative models for many are reviewed by Neubig et al (2003). A simple pharmacologic response measured may be perceived as the result of many receptors and biological processes working together. The whole series of events may be the interplay of (1) pharmacologic response due to drug–receptor interaction, (2) physiologic or compensatory action or adjustment by the body, (3) pharmacogenetics that modify the drug response, and (4) alteration in biological responses due to an intervention by a pathogenic process. A comprehensive model incorporating all aspects is ideally best but may be too complicated to measure and validate in practice. An alternative approach is the monitoring of a biologic marker(s) in which the events in the biological processes may be observed directly or indirectly. The general approach for applying markers or biomarkers in pharmacodynamics is deeply rooted in pharmacology and pharmacokinetics and is illustrated by the modeling of various drugs shown in this chapter. Biomarkers broadly include markers that characterize the disease and the physical/biological changes associated with its progress in or drug treatment response in the body. During drug development, biomarkers developed with appropriate PD models can greatly increase understanding of drug therapy as well as the underlying disease.

Unlike pharmacokinetic modeling, pharmacodynamic modeling has a greater advantage in that the clinical response (eg, change in blood pressure or clotting time) is affected by disposition of the drug as well as many events at the receptor site that may modulate actual pharmacologic action. In a simple case, drug that is systemically absorbed is transported to the site of action (receptor site) in which the pharmacologic receptor resides. Drug–receptor binding may occur, causing a response directly or through signal transduction that leads to the response. Observation of a drug response may involve several biologic steps as listed above and therefore, pharmacodynamic modeling must quantitatively account for all the processes involved to be useful.

Pharmacodynamic (PD) models involve complex mechanisms that may not be easily simplified. Earlier PD models in this chapter are empirical models since the pharmacodynamic mechanisms were not well understood. Empirical models may help eliminate an unlikely pharmacokinetic mechanism. Generally, PK–PD models should be mechanism based whenever possible for simulations to be predictive. The understanding of drug response is greatly enhanced when pharmacokinetic modeling techniques are combined with clinical pharmacology, resulting in the development of mechanism-based pharmacokinetic–pharmacodynamic models. Pharmacokinetic–pharmacodynamic models use data derived from the plasma drug concentration-versus-time profile and from the time course of the pharmacologic effect to predict the pharmacodynamics of the drug. Pharmacokinetic–pharmacodynamic models have been reported for virtually every category of drugs. Examples in this chapter include antipsychotics, anticoagulants, neuromuscular blockers, antihypertensives, antiinfectives, and sedatives.

The relationship between drug response and pharmacokinetics of many drugs may be explained in simple models. The computer model–simulated response based on assumptions should be distinguished from those models that are based on results obtained from clinical trials. These models are far more sophisticated and objective. Clinical effects in drug development are usually well defined, multicentered, and may be performed with special patient inclusion/exclusion criteria, subject size, controls, and other considerations as described in the clinical study protocols. The FDA publishes regulatory guidances that describe various aspects of e-study design and data analysis. Pharmacodynamics is indispensible for understanding drug receptors and disease state variables. Disease progress and drug therapy are both intimately related to pharmacokinetics (Chapter 12).

Biomarkers (BM), Pharmacodynamics (PD), and Clinical Endpoints (CE)

Biomarkers have been used to monitor biologic or genetic changes associated with the progress of a disease or a pharmacologic action.1 Biomarkers may include gene variants, functional deficiencies, expression changes, chromosomal abnormalities, and others. For pharmacodynamic modeling of a drug, the biomarker (BM) selected should ideally indicate the biological/pathological processe(es) and/or pharmacological response(s) modified due to the therapeutic intervention.

A biomarker is measured objectively and specifically evaluated as an indicator of normal biologic processes, pathogenic processes, or biological responses to a therapeutic intervention. A biomarker can also define a physiologic, pathologic, or anatomic characteristic or measurement that is thought to relate to some aspect of normal or abnormal biologic function. Data from genomic and proteomics differentiating healthy versus disease states can therefore lead to the discovery of new biomarkers. Rational use of BMs can accelerate drug development and decision making and can provide a bridge between mechanistic preclinical studies and empiric clinical testing. The development of BM science including its definition has evolved during the last two decades (Lesko and Atkinson, 2001; Rolan et al 2003; Colburn and Lee, 2003). The use of BMs may be defined and developed by different disciplines to address special needs (FDA draft guidance, 2011).

During drug development, clinical data and special markers also may be developed to demonstrate safety and efficacy. Clinical endpoints (CE) measure how patients feel, function, or survive. Since clinical endpoint data may be somewhat subjective, the CE (eg, pain) is often categorized (eg, using a 0–5 scale) to reflect the intensity of the feeling for quantitation. These clinical observations or treatment results may be partly or totally related to a BM. The surrogate markers (SM) or surrogate endpoints may be more objective and are designed to replace CE for efficacy and/or toxicity under a set of well-defined criteria. Clinically, few biomarkers are developed as surrogate endpoints due to the complexity of the disease. For example, the surrogate endpoints blood pressure and serum cholesterol concentrations may be used in evaluation of cardiovascular drugs (Temple, 1999). For example, the most widely used surrogate biomarkers are plasma concentrations of drugs that reflect systemic exposure to drug including bioavailability, bioequivalence, and pharmacokinetic guide for dosage regimens in clinical practice (eg, therapeutic drug monitoring).

Biomarkers may reduce uncertainty in drug development and evaluation by providing quantitative predictions about drug performance, and with PK–PD modeling simulation, can play a critical role in the drug development process. Biomarkers are broadly used to follow disease or drug treatment effects over time and thus are important for understanding drug mechanism (Lesko and Atkinson 2001; Rolan et al, 2003). However, because the range of biologic measurements that can be considered biomarkers is now so broad, some classification and stratification are needed to provide an understanding of what types of biologic measurements are used for which purposes. The following mechanistic classification of BMs was cited by the International committee on BM (Rolan et al, 2003) (Table 19-1).

Clinical Considerations in the Use of Biomarkers

Most biomarkers are endogenous macromolecules, which are measured in vivo in biological fluids. However, not all biomarkers reflect in vivo processes (Stern et al, 2003). To be most informative in drug development, a biomarker assay or assays should measure in vivo drug effects, not drug concentrations. Stern et al (2003) cited an example involving the clotting time and activated partial thromboplastin time measured for ecarin.2 These coagulation tests that seemingly meet the definition of a biomarker by prolongation of the coagulation times closely correlated with blood concentrations of the oral thrombin inhibitor. However, these tests reflect enzyme inhibition assays as a function of drug concentration. Changes in coagulation test results demonstrate only that ex vivo clot formation has been altered but do not indicate that an in vivo process has been affected. Reliable and selective assays could be validated under a GLP-like environment for quantitative methods (Colburn and Lee, 2003).

In early-phase clinical drug development, biomarkers may be used to guide dose selection and escalation. For a few drugs, well-characterized pharmacokinetic–pharmacodynamic (PK–PD) relationships can support, without further clinical data, therapeutic potential in a new target population, or justification for a different formulation or dosing regimen. Several considerations are important for BM selection to improve drug development based on Colburn and Lee (2003): (1) mechanism-based biomarker selection and correlation to clinical endpoints; (2) quantification of drug and/or metabolites in biological fluids under good laboratory practices (GLP); (3) GLP-like biomarker method validation and measurements; and (4) mechanism-based PK/PD modeling and validation.

Changes in biomarkers following drug treatment may predict or identify safety problems related to a drug candidate or reveal a pharmacological activity expected to predict an eventual benefit from treatment. These measurements may be related to the mechanism of response to treatments or may be useful to evaluate the therapeutic response or clinical benefit endpoints. According to some scientists (Temple, 1999; Lesko and Atkinson, 2001), biomarkers:

• May be valid surrogates for clinical benefit (eg, blood pressure, cholesterol, viral load)
• May reflect the pathologic process and be at least candidate surrogates (eg, brain appearance in Alzheimer's disease, brain infarct size, various radiographic/isotopic function tests)
• Reflect drug action but are of uncertain relation to clinical outcome (eg, inhibition of ADP-dependent platelet aggregation, ACE inhibition)

A pharmacodynamic biomarker may be described as a dynamic assessment that shows that a biological response has occurred in a patient after receiving a drug for treatment. A pharmacodynamic biomarker may be treatment-specific or more broadly informative of disease response. Examples include blood pressure, cholesterol, HbA1C, intraocular pressure, radiographic measures, and C-reactive protein. However, even if carefully chosen, BMs may fail to become surrogate endpoints.

Pharmacogenomic Biomarkers in Drug Labels

Pharmacogenomics can play an important role in identifying responders and non-responders to medications, avoiding adverse events, and optimizing drug dose. Drug labels (FDA 2011) may contain information on genomic biomarkers and can describe:

• Drug exposure and clinical response variability
• Risk for adverse events
• Genotype-specific dosing
• Mechanisms of drug action
• Polymorphic drug target and disposition genes

Drug Receptors and the Development of Pharmacokinetic–Pharmacodynamic (PK–PD) Models

The description of drug receptors was historically based on observations of drug response using known drug agonists and antagonists. From both these drug response observations and from pharmacokinetic data, PK–PD models were developed to describe quantitative relationships. Advances in molecular biology now allow the molecular structures of many drug receptors to be characterized and their locations elucidated (Katzung et al, 2009). Receptors are generally protein or macromolecules located inside or outside cell membranes. A tabulation of various receptors is listed in Table 19-2.

Receptors are responsible for selectivity of drug action. The nature of the receptors determines the quantitative relationship between the drug dose or drug concentration and the pharmacologic effect. This relationship is an important basis for PD. The receptor's affinity for drug binding and the total number of receptors available determine the concentration of drug required to exert a PD response and the number of drug–receptor complexes which may limit the attainment of a maximal effect. This relationship forms the basis of the Emax model discussed in later sections of this chapter.

The molecular size, shape, and electrical charge of a drug determine whether a drug will bind to a particular receptor among the vast number of chemically and structurally different binding sites in a cell, tissue, or patient. The drug may be referred to as a ligand when the drug binds a macromolecule or as a substrate when the receptor is an enzyme. An adverse drug reaction can occur when a drug interacts with different, unintended receptors, thus affecting therapeutic efficacy and toxic effects. The drug's relationship to drug receptor binding explains drug classes such as agonists, partial agonists, and antagonists. Elucidating this relationship also greatly helps the development of new drug therapies. Drugs may be designed to mimic or reproduce natural ligands, such as hormones and neurotransmitters.

Pharmacokinetic–Pharmacodynamic (PK–PD) Models

As discussed, PK–PD models relate the pharmacological effect of a drug to the log concentration of the drug in the plasma or in other body fluids close to the receptor site. The model equations created predict the time course of drug action and help to more accurately estimate a therapeutic dosage regimen for the patient. Similarly, the PK–PD model may relate drug concentrations to side effects or adverse events, such as the reduction of QT interval or slowing of repolarization of the heart due to side effects of some drugs such as torsades de pointes. The association between torsades and a prolonged QT interval has long been known. Common drugs such as erythromycin, clarithromycin, terfenadine, and some antihistamines may cause prolongation of the QT interval without necessarily causing overt clinical problems. PK–PD models can provide a much better understanding of how the mechanism of the drug acts and improves safety of these drugs. For either new or newly approved drugs, clinical information may be limited until adequate Phase 4 studies are completed. Phase 4 post marketing studies provide more information on the incidence of side effects and drug efficacy in various subpopulations. Mechanistic studies of a drug often reveal that there are biological changes as measured using biomarkers in the body associated with drug use.

During PK–PD modeling, it is important to describe the following prospectively:

1. Statement of the Problem

   The objectives of modeling, the study design, and the available PK and PD data

2. Statement of Assumptions

   The assumptions of the model regarding dose-response, PK, PD, and/or one or more of the following:

   • The mechanism of the drug actions for efficacy and adverse effects
   • Immediate or cumulative clinical effects
   • Development or absence of tolerance
   • Drug-induced inhibition or induction of PK processes
   • Disease state progression
   • Response in a placebo group

1A genomic biomarker is a measurable DNA and/or RNA characteristic that is an indicator of normal biologic processes, pathogenic processes, and/or response to therapeutic or other interventions (FDA guidance, 2008).

2Ecarin is a metalloproteinase from snake venom that activates prothrombin to meizothrombin.

The onset, intensity, and duration of the pharmacologic effect depend on the dose and the pharmacokinetics of the drug. As the dose increases, the drug concentration at the receptor site increases, and the pharmacologic response (effect) increases up to a maximum effect. A plot of the pharmacologic effect to dose on a linear scale generally results in a hyperbolic curve with maximum effect at the plateau (Fig. 19-1). The same data may be compressed and plotted on a log–linear scale and results in a sigmoid curve (Fig. 19-2).

For many drugs, the graph of the log dose–response curve shows a linear relationship at a dose range between 20% and 80% of the maximum response, which typically includes the therapeutic dose range for many drugs. For a drug that follows one-compartment pharmacokinetics, the volume of distribution is constant; therefore, the pharmacologic response is also proportional to the log plasma drug concentration within a therapeutic range, as shown in Fig. 19-3.

Mathematically, the relationship in Fig. 19-3 may be expressed by the following equation, where m is the slope, e is an extrapolated intercept, and E is the drug effect at drug concentration C:

Solving for log C yields

However, after an intravenous dose, the concentration of a drug in the body in a one-compartment open model is described as follows:

By substituting Equation 19.2 into Equation 19.3, we get Equation 19.4, where E0 = effect at concentration C0:

The theoretical pharmacologic response at any time after an intravenous dose of a drug may be calculated using Equation 19.4. Equation 19.4 predicts that the pharmacologic effect will decline linearly with time for a drug that follows a one-compartment model, with a linear log dose–pharmacologic response. From this equation, the pharmacologic effect declines with a slope of km/2.3. The decrease in pharmacologic effect is affected by both the elimination constant k and the slope m. For a drug with a large m, the pharmacologic response declines rapidly and multiple doses must be given at short intervals to maintain the pharmacologic effect.

The relationship between pharmacokinetics and pharmacologic response can be demonstrated by observing the percent depression of muscular activity after an IV dose of ± tubocurarine. The decline of pharmacologic effect is linear as a function of time (Fig. 19-4). For each dose and resulting pharmacologic response, the slope of each curve is the same. Because the values for each slope, which include km (Equation 19.4), are the same, the sensitivity of the receptors for ± tubocurarine is assumed to be the same at each site of action. Note that a plot of the log concentration of drug versus time yields a straight line.

A second example of the pharmacologic effect declining linearly with time was observed with lysergic acid diethylamide, or LSD (Fig. 19-5). After an IV dose of the drug, log concentrations of drug decreased linearly with time except for a brief distribution period. Furthermore, the pharmacologic effect, as measured by the performance score of each subject, also declined linearly with time. Because the slope is governed in part by the elimination rate constant, the pharmacologic effect declines much more rapidly when the elimination rate constant is increased as a result of increased metabolism or renal excretion. Conversely, a longer pharmacologic response is experienced in patients when the drug has a longer half-life.

The relationship between the duration of the pharmacologic effect and the dose can be inferred from Equation 19.3. After an intravenous dose, assuming a one-compartment model, the time needed for any drug to decline to a concentration C is given by the following equation, assuming the drug takes effect immediately:

Using Ceff to represent the minimum effective drug concentration, the duration of drug action can be obtained as follows:

Some practical applications are suggested by this equation. For example, a doubling of the dose will not result in a doubling of the effective duration of pharmacologic action. On the other hand, a doubling of t1/2 or a corresponding decrease in k will result in a proportional increase in duration of action. A clinical situation is often encountered in the treatment of infections in which Ceff is the bacteriocidal concentration of the drug, and, in order to double the duration of the antibiotic, a considerably greater increase than simply doubling the dose is necessary.

The minimum effective concentration (MEC or Ceff) in plasma for a certain antibiotic is 0.1 μg/mL. The drug follows a one-compartment open model and has an apparent volume of distribution, VD, of 10 L and a first-order elimination rate constant of 1.0 h–1.

1. What is the teff for a single 100-mg IV dose of this antibiotic?

2. What is the new teff or t′eff for this drug if the dose were increased 10-fold, to 1000 mg?

Solution

1. The teff for a 100-mg dose is calculated as follows. Because VD = 10,000 mL,

   

   For a one-compartment-model IV dose, C = C0e–kt. Then,

   

2. The t′eff for a 1000-mg dose is calculated as follows (prime refers to a new dose). Because VD = 10,000 mL,

   

   and

   

   The percent increase in teff is therefore found as

   Percent increase in

   Precent increase in

   Percent increase in teff = 50%

This example shows that a 10-fold increase in the dose increases the duration of action of a drug (teff) by only 50%.

A single equation can be derived to describe the relationship of dose (D0) and the elimination half-life (t1/2) on the effective time for therapeutic activity (teff). This expression is derived below.

ln Ceff = ln C0 − kteff

Because C0 = D0/VD,

Substituting 0.693/t1/2 for k,

From Equation 19.8, an increase in t1/2 will increase the teff in direct proportion. However, an increase in the dose, D0, does not increase the teff in direct proportion. The effect of an increase in VD or Ceff can be seen by using generated data. Only the positive solutions for Equation 19.8 are valid, although mathematically a negative teff can be obtained by increasing Ceff or VD. The effect of changing dose on teff is shown in Fig. 19-6 using data generated with Equation 19.8. A nonlinear increase in teff is observed as dose increases.

Because elimination of drugs is due to the processes of excretion and metabolism, an alteration of any of these elimination processes will affect the t1/2 of the drug. In certain disease states, pathophysiologic changes in hepatic or renal function will decrease the elimination of a drug, as observed by a prolonged t1/2. This prolonged t1/2 will lead to retention of the drug in the body, thereby increasing the duration of activity of the drug (teff) as well as increasing the possibility of drug toxicity.

To improve antibiotic therapy with the penicillin and cephalosporin antibiotics, clinicians have intentionally prolonged the elimination of these drugs by giving a second drug, probenecid, which competitively inhibits renal excretion of the antibiotic. This approach to prolonging the duration of activity of antibiotics that are rapidly excreted through the kidney has been used successfully for a number of years. Similarly, Augmentin is a combination of amoxicillin and clavulanic acid; the latter is an inhibitor of β-lactamase. This β-lactamase is a bacterial enzyme that degrades penicillin-like drugs. The data in Table 19-3 illustrate how a change in the elimination t1/2 will affect the teff for a drug. For all doses, a 100% increase in the t1/2 will result in a 100% increase in the teff. For example, for a drug whose t1/2 is 0.75 hour and that is given at a dose of 2 mg/kg, the teff is 3.24 hours. If the t1/2 is increased to 1.5 hours, the teff is increased to 6.48 hours, an increase of 100%. However, the effect of doubling the dose from 2 to 4 mg/kg (no change in elimination processes) will only increase the teff to 3.98 hours, an increase of 22.8%. The effect of prolonging the elimination half-life has an extremely important effect on the treatment of infections, particularly in patients with high metabolism, or clearance, of the antibiotic. Therefore, antibiotics must be dosed with full consideration of the effect of alteration of the t1/2 on the teff. Consequently, a simple proportional increase in dose will leave the patient's blood concentration below the effective antibiotic level most of the time during drug therapy. The effect of a prolonged teff is shown in lines a and c in Fig. 19-7, and the disproportionate increase in teff as the dose is increased 10-fold is shown in lines a and b.

Pharmacokinetic/Pharmacodynamic Relationships and Efficacy of Antibiotics

In the previous section, the time above the effective concentration, teff, was shown to be important in optimizing the therapeutic response of many drugs. This concept has been applied to antibiotic drugs (Drusano, 1988; Craig, 1995; Craig and Andes, 1996; Scaglione, 1997). For example, Craig and Andes (1996) discussed the antibacterial treatment of otitis media. Using the minimum inhibitory antibiotic concentration (MIC) for the microorganism in serum, the percent time for the antibiotic drug concentration to be above the MIC was calculated for several antibacterial classes, including cephalosporins, macrolides, and trimethoprim-sulfamethoxazole (TMP/SMX) combination (Table 19-4). Although the drug concentration in the middle ear fluid (MEF) is important, once the ratio (MEF/serum) is known, the serum drug level may be used to project MEF drug levels. The percent time above MIC of the dosing interval during therapy correlated well to the percent of bacteriologic cure (Fig. 19-8). An almost 100% cure was attained by maintaining the drug concentration above the MIC for 60%–70% of the dosing interval; an 80%–85% cure was achieved with 40%–50% of the dosing interval above MIC. When the percent of time above MIC falls below a critical value, bacteria will regrow, thereby prolonging the time for eradication of the infection. The pharmacokinetic model was further supported by experiments from a mouse infection model in which an infection in the thigh due to Pseudomonas aeruginosa was treated with ticarcillin and tobramycin.

In another study, Craig (1995) compared the AUC/MIC, the time above MIC, and drug peak concentration over MIC and found that the best fit was obtained when colony-forming units (CFUs) were plotted versus time above MIC for cefotaxime in a mouse infection model (Fig. 19-9).

Both Drusano (1988) and Craig (1995) reviewed the relationship of pharmacokinetics and pharmacodynamics in the therapeutic efficacy of antibiotics. For some antibiotics, such as the aminoglycosides and fluoroquinolones, both the drug concentration and the dosing interval have an influence on the antibacterial effect. For some antibiotics, such as the β-lactams, vancomycin, and the macrolides, the duration of exposure (time-dependent killing) or the time the drug levels are maintained above the MIC (teff) is most important for efficacy. For many antibiotics (eg, fluoroquinolones), there is a defined period of bacterial growth suppression after short exposures to the antibiotic. This phenomenon is known as the postantibiotic effect (PAE). Other influences on antibiotic activity include the presence of active metabolite(s), plasma drug protein binding, and the penetration of the antibiotic into the tissues. In addition, the MIC for the antibiotic depends on the infectious microorganism and the resistance of the microorganism to the antibiotic. In the case of ciprofloxacin, a quinolone, the percent of cure of infection at various doses was better related to AUIC, which is the product of area under the curve and the reciprocal of minimum inhibition concentration, MIC (Forrest et al, 1993). Interestingly, quinolones inhibit bacterial DNA gyrase, quite different from the β-lactam antibiotics, which involve damage to bacterial cell walls.

Relationship between Systemic Exposure and Response—Anticancer Drugs

Plasma drug concentrations for drugs that have highly variable drug clearance in patients fluctuate widely even after intravenous infusion (Rodman and Evans, 1991). For highly variable drugs, there is no apparent relationship between the therapeutic response and the drug dose. For example, the anticancer drug teniposide at three different doses give highly variable steady-state drug concentrations and therapeutic response (Fig. 19-10). In some patients, single-point drug concentrations were variable and even higher with lower doses. Careful pharmacokinetic–pharmacodynamic analysis showed that a graded response curve may be obtained when responses are plotted versus systemic exposure as measured by “concentration × time” which is an integration of area under the concentration curve. For example, the response of quinolones is often related to AUC/MIC. Figure 19-11 demonstrates another example showing that anticancer response may be better correlated to total area under the drug concentration curve (AUC), even when no apparent dose–response relationship is observed. Undoubtedly, the cytotoxic effect of the drug involves killing cancer cells with multiple-resistance thresholds that require different time exposures to the drug. The objective of applying pharmacokinetic–pharmacodynamic principles is to achieve therapeutic efficacy without triggering drug toxicity. This relationship is illustrated by the sigmoid curves for response and toxicity (see Fig. 19-11), each of which is close to the other and intensify as drug concentration increases.

The rate of drug absorption influences the rate in which the drug gets to the receptor and the subsequent pharmacologic effect. For drugs that exert an acute pharmacologic effect, usually direct-acting drug agonists, extremely rapid drug absorption may have an intense and possibly detrimental effect. For example, niacin (nicotinic acid) is a vitamin given in large doses to decrease elevated plasma cholesterol and triglycerides. Rapid systemic absorption of niacin when given in an immediate-release tablet will cause vasodilation, leading to flushing and postural hypertension. Extended-release niacin products are preferred because the more slowly absorbed niacin allows the baroreceptors to adjust to the vasodilation and hypotensive effects of the drug. Phenylpropanolamine was commonly used as a nasal decongestant in cough and cold products or as an anorectant in weight-loss products. Phenylpropanolamine acts as a pressor, increasing the blood pressure much more intensely when given as an immediate-release product compared to an extended-release product.

Equilibration Pharmacodynamic Half-Life

For some drugs, the half-time for drug equilibration has been estimated by observing the onset of response. A list of drug half-times reported by Lalonde (1992) is shown in Table 19-5. The factors that affect this parameter include perfusion of the effect compartment, blood–tissue partitioning, drug diffusion from capillaries to the effect compartment, protein binding, and elimination of the drug from the effect compartment.

Substance Abuse Potential

The rate of drug absorption has been associated with the potential for substance abuse. Drugs taken by the oral route have the lowest abuse potential. For example, cocoa leaves containing cocaine alkaloid have been chewed by South American Indians for centuries (Johanson and Fischman, 1989). Cocaine abuse has become a problem as a result of the availability of cocaine alkaloid (“crack” cocaine) and because of the use of other routes of drug administration (intravenous, intranasal, or smoking) that allow a very rapid rate of drug absorption and onset of action (Cone, 1995). Studies on diazepam (deWit et al, 1993) and nicotine (Henningfield and Keenan, 1993) have shown that the rate of drug delivery correlates with the abuse liability of such drugs. Thus, the rate of drug absorption influences the abuse potential of these drugs, and the route of drug administration that provides faster absorption and more rapid onset leads to greater abuse.

The study of drug tolerance and physical dependency is of particular interest in understanding the actions of abused drug substances, such as opiates and cocaine. Drug tolerance is a quantitative change in the sensitivity of the drug of the receptor site and is demonstrated by a decrease in pharmacodynamic effect after repeated exposure to the same drug. The degree of tolerance may vary greatly (Cox, 1990). Drug tolerance has been well described for organic nitrates, opioids, and other drugs. For example, the nitrates relax vascular smooth muscle and have been used for both acute angina (eg, nitroglycerin sublingual spray or transmucosal tablet) or angina prophylaxis (eg, nitroglycerin transdermal, oral controlled- release isosorbide dinitrate). Well-controlled clinical studies have shown that tolerance to the vascular and antianginal effects of nitrates may develop. For nitrate therapy, the use of a low nitrate or nitrate-free periods has been advocated as part of the therapeutic approach. The magnitude of drug tolerance is a function of both the dosage and the frequency of drug administration. Cross tolerance can occur for similar drugs that act on the same receptors. Tolerance does not develop uniformly to all the pharmacologic or toxic actions of the drug. For example, patients who show tolerance to the depressant activity of high doses of opiates will still exhibit “pinpoint” pupils and constipation.

The mechanism of drug tolerance may be due to (1) disposition or pharmacokinetic tolerance or (2) pharmacodynamic tolerance. Pharmacokinetic tolerance is often due to enzyme induction (discussed in earlier chapters), in which the hepatic drug clearance increases with repeated drug exposure. Pharmacodynamic tolerance is due to a cellular or receptor alteration in which the drug response is less than what is predicted in the patient given subsequent drug doses. Measurement of serum drug concentrations may differentiate between pharmacokinetic tolerance and pharmacodynamic tolerance. Acute tolerance, or tachyphylaxis, which is the rapid development of tolerance, may occur due to a change in the sensitivity of the receptor or depletion of a cofactor after only a single or a few doses of the drug. Drugs that work indirectly by releasing norepinephrine may show tachyphylaxis. Drug tolerance should be differentiated from genetic factors which account for normal variability in the drug response.

Physical dependency is demonstrated by the appearance of withdrawal symptoms after cessation of the drug. Workers exposed to volatile organic nitrates in the workplace may initially develop headaches and dizziness followed by tolerance with continuous exposure. However, after leaving the workplace for a few days, the workers may demonstrate nitrate withdrawal symptoms. Factors that may affect drug dependency may include the dose or amount of drug used (intensity of drug effect), the duration of drug use (months, years, and peak use), and the total dose (amount of drug × duration). The appearance of withdrawal symptoms may be abruptly precipitated in opiate-dependent subjects by the administration of naloxone (Narcan), an opioid antagonist that has no agonist properties.

Many drug responses, such as hypersensitivity and allergic responses, are not fully explained by pharmacodynamics and pharmacokinetics. Allergic responses generally are not dose related, although some penicillin-sensitive patients may respond to threshold skin concentrations, but otherwise no dose–response relationship has been established. Skin eruption is a common symptom of drug allergy. Allergic reactions can occur at extremely low drug concentrations. Some urticaria episodes in patients have been traced to penicillin contamination in food or to penicillin contamination during dispensing or manufacturing of other drugs. A patient's allergic reactions are important data that must be recorded in the patient's profile along with other adverse reactions. Penicillin allergic reaction in the population is often detected by skin test with benzylpenicilloyl polylysine (PPL). The incidence of penicillin allergic reaction occurs in about 1%–10% of patients. The majority of these reactions are minor cutaneous reactions such as urticaria, angioedema, and pruritus. Serious allergic reactions such as anaphylaxis are rare, with an incidence of 0.021%–0.106% for penicillins (Lin, 1992). For cephalosporins, the incidence of anaphylactic reaction is less than 0.02%. Anaphylactic reaction for cefaclor was reported to be 0.001% in a postmarketing survey. There are emerging trends showing that there may be a difference between the original and the new generations of cephalosporins (Reisman and Reisman, 1995). Cross-sensitivity to similar chemical classes of drugs can occur.

Allergic reactions may be immediate or delayed and have been related to IgE mechanisms. In β-lactam (penicillin) drug allergy, immediate reactions occur in about 30 to 60 minutes, but either a delayed reaction or accelerated reaction may occur from 1 to 72 hours after administration. Anaphylactic reaction may occur in both groups. Although some early evidence of cross-hypersensitivity between penicillin and cephalosporin was observed, the incidence in patients sensitive to penicillin shows only a twofold increase in sensitivity to cephalosporin compared with that of the general population. The report rationalized that it is safe to administer cephalosporin to penicillin-sensitive patients and that the penicillin skin test is not useful in identifying patients who are allergic to cephalosporin, because of the low incidence of cross-reactivity (Reisman and Reisman, 1995). In practice, the clinician should evaluate the risk of drug allergy against the choice of alternative medication. Some earlier reports showed that cross-sensitivity between penicillin and cephalosporin was due to the presence of trace penicillin present in cephalosporin products.

After systemic absorption, the drug is carried throughout the body by the general circulation. Most of the drug dose will reach unintended target tissues, in which the drug may be passively stored, produce an adverse effect, or be eliminated. A fraction of the dose will reach the target site and combine with the receptor which is generally located in the tissue. The receptor site is unknown most of the time, but, kinetically, it is known as the effect compartment. The time course of drug delivery to the effect compartment will determine whether the onset of pharmacologic response is immediate or delayed. The delivery of drug to the effect compartment is affected by the rate of blood flow, diffusion, and partition properties of the drug and the receptor molecules.

At the receptor site, the onset, duration, and intensity of the pharmacologic response are controlled by receptor concentration and the concentration of the drug and/or its active metabolites. The ultimate pharmacologic response (effect) may depend largely on the stereospecific nature of the interaction of the drug with the receptor and the rates of association and dissociation of the drug–receptor complex. Depending on their location and topography, not all receptor molecules are occupied by drug molecules when a maximum pharmacologic response is produced. Other variables, such as age, sex, genetics, nutrition, and tolerance, may also modify the pharmacologic response, making it difficult to relate the pharmacologic response to plasma drug concentration. To control data fluctuation and simplify pharmacodynamic fitting, the pharmacologic response is often expressed as a percent of response above a baseline or percent of maximum response. By combining pharmacokinetics and pharmacodynamics, some drugs with relatively complex pharmacologic responses have been described by pharmacodynamic models that account for their onset, intensity, and duration of action.

After the pharmacodynamics of a drug are characterized, the time course of pharmacologic response may be predicted after drug administration. Also, from these data, it is possible to determine from the pharmacokinetic parameters whether an observed change in pharmacologic response is due to pharmacodynamic factors, such as tachyphylaxis or tolerance, or due to pharmacokinetic factors, such as a change in drug absorption, elimination, or distribution.

Drug–Receptor Theory Relating Pharmacologic Effect and Dose

The relationship between pharmacologic effect and dose was advanced by Wagner (1968), who derived a kinetic expression that relates drug concentration to pharmacologic effect. This theoretical development transformed the semi-empirical dose–effect relationship (the hyperbolic or log sigmoid profile) into a theoretical equation that relates pharmacologic effect to pharmacokinetics (ie, a pharmacokinetics/pharmacodynamic, PK/PD model). Because the equation was developed for a drug receptor with either single or multiple drug binding, many drugs with a sigmoid concentration effect profile may be described by this model. The slope of the profile also provides some insight into the drug–receptor interaction.

The basic equation mimics somewhat the kinetic equation for protein drug binding (Chapter 10). One or more drug molecules may interact with a receptor to form a complex that in turn elicits a pharmacodynamic response, as illustrated in Fig. 19-12. The rate of change in the number of drug–receptor complexes is expressed as db/dt. From Fig. 19-12, a differential equation is obtained as shown:

where k1cs (a – b) is rate of receptor complex formation and bk2 = rate of dissociation of the receptor complex.

At steady state, db/dt = 0 and Equation 19.9 reduces to

For many drugs, the pharmacologic response (R) is proportional to the number of receptors occupied:

The pharmacologic response (R) is related to the maximum pharmacologic response (Rmax), concentration of drug, and rate of change in the number of drug receptor complexes occupied:

A graph of Equation 19.12 constructed from the percent pharmacologic response, (R/Rmax) × 100, versus the concentration of drug gives the response–concentration curve (Fig. 19-13). This type of theoretical development explains that the pharmacologic response–dose curve is not completely linear over the entire dosage range, as is frequently observed.

The total pharmacologic response elicited by a drug is difficult to quantitate in terms of the intensity and the duration of the drug response. The integrated pharmacologic response is a measure of the total pharmacologic response and is expressed mathematically as the product of these two factors (ie, duration and intensity of drug action) summed up over a period of time. Using Equation 19.12, an integrated pharmacologic response is generated if the drug plasma concentration–time curve can be adequately described by a pharmacokinetic model.

Table 19-6 is based on a hypothetical drug that follows a one-compartment open model. The drug is given intravenously in divided doses. With this drug, the total integrated response increases considerably when the total dose is given in a greater number of divided doses. By giving the drug in a single dose, two doses, four doses, and eight doses, an integrated response was obtained that ranged from 100% to 138.9%, using the single-dose response as a 100% reference. It should be noted that when the bolus dose is broken into a smaller number of doses, the largest percent increase in the integrated response occurs when the bolus dose is divided into two doses. Further division will cause less of an increase, proportionally. The actual percent increase in integrated response depends on the t1/2 of the drug as well as the dosing interval.

The values in Table 19-6 were generated from theory. However, these data illustrate that the pharmacologic response depends on the dosing schedule. A large total dose given in divided doses may produce a pharmacologic response quite different from that obtained by administering the drug in a single dose.

Correlation of pharmacologic response to pharmacokinetics is not always possible with all drugs. Sometimes intermediate steps are involved in the mechanism of drug action that are more complex than is assumed in the model. For example, warfarin (an anticoagulant) produces a delayed response, and there is no direct correlation of the anticoagulant activity to the plasma drug concentration. The plasma warfarin level is correlated with the inhibition of the prothrombin complex production rate. However, many correlations between pharmacologic effect and plasma drug concentration are performed by proposing models that may be discarded after more data are collected. The process of pharmacokinetic modeling can greatly enhance our understanding of the way drugs act in a quantitative manner.

No unified general pharmacodynamic model based on detailed drug–receptor theory that relates pharmacologic response to pharmacokinetics is available. Most of the drug–receptor-based models are descriptive and lack quantitative details. Successful modeling of pharmacologic response has been achieved with semi-empirically based assumptions and usually with some oversimplification of the real process. Many of the classic pharmacodynamic models were developed without detailed knowledge of the drug–receptor interaction. The successful modeling of the degree of muscle paralysis of d-tubocurarine to plasma concentrations is an interesting example in which the exact mechanism of the drug–receptor interaction was not considered. One of the few pharmacodynamic models that takes into account the interaction between the receptor and the drug molecule leading to a pharmacologic effect was described by Boudinot et al (1986) using the drug prednisolone as an example. Prednisolone is a corticosteroid that binds to cytosolic receptors within the cell (Fig. 19-14). The bound steroid receptor complex is activated and translocated into the nucleus of the cell. Within the cell, the drug–receptor complex associates with specific DNA sequences and modulates the transcription of RNA, which ultimately initiates protein synthesis (Boudinot et al, 1986). Tyrosine aminotransferase (TAT) is an enzyme protein that is increased (induced) by the action of prednisolone. In the liver cell, the prednisolone concentration, drug–receptor concentration, and TAT enzyme were measured with respect to time.

The pharmacodynamic model accounted for the delayed response of prednisolone, a characteristic of corticosteroid response. In this model, prednisolone is first bound to plasma protein, and free drug must leave the plasma compartment and enter the cell to form a drug–receptor complex; creation of this complex then triggers the pharmacologic events leading to an increase in intracellular TAT concentration. A decrease in free receptor or an increase in bound receptor complexes after drug administration was observed. Plasma prednisolone concentrations were described by a triexponential equation, and a time lag was built into the model to account for the delay between TAT increase and the drug–receptor–DNA complex formation (Figs. 19-15 and 19-16). A review of PK–PD modelling has been published by Meibohm and Derendorf (1997).

Equations Used in Pharmacodynamic Models

Historically, pharmacologists have noted that pharmacodynamic effects increase with increasing drug concentration and then reach a plateau or maximum pharmacologic effect. With application in many fields, the Hill equation was developed in 1910 (Goutelle et al, 2008; Weiss, 1997), later used to describe nonlinear and saturable mechanisms, such as the binding of oxygen to hemoglobin. The Hill equation has been used as an empirical equation to describe the relationship between drug concentration and pharmacologic response (Equation 19.12b). The better known sigmoid curve (or S-curve) is the logistic function, Equation 19.12a. The two equations are mathematically related as shown below. The logistic function is defined by the equation where x is the independent variable and a and b are constants:

If a is redefined as Ln(Kb), and x as Ln z, then the Hill equation is obtained, which looks more similar to equation 19.14a that is used in Emax models.

Maximum Effect (Emax) Model

The maximum effect model (Emax) is an empirical model that relates pharmacologic response to drug concentrations. This model incorporates the observation known as the law of diminishing return, which shows that an increase in drug concentration near the maximum pharmacologic response produces a disproportionately smaller increase in the pharmacologic response (Fig. 19-17). The Emax model describes drug action in terms of maximum effect (Emax) and EC50, the drug concentration that produces 50% maximum pharmacologic effect.

where C is the plasma drug concentration and E is the pharmacologic effect.

Equation 19.13 is a saturable process resembling Michaelis–Menton enzyme kinetics. As the plasma drug concentration C increases, the pharmacologic effect E approaches Emax asymptotically. A double-reciprocal plot of Equation 19.13 may be used to linearize the relationship, similar to a Lineweaver–Burke equation.

Emax is the maximum pharmacologic effect that may be obtained by the drug. EC50 is the drug concentration that produces one-half (50%) of the maximum pharmacologic response. In this model, both Emax and EC50 can be measured. For example, the bronchodilator activity of theophylline may be monitored by measuring FEV1 (forced expiratory volume) at various plasma drug concentrations (Fig. 19-18). For theophylline, a small gradual increase in FEV1 is obtained as the plasma drug concentrations are increased higher than 10 mg/L. Only a 17% increase in FEV1 is observed when the plasma theophylline concentration is doubled from 10 to 20 mg/L. The EC50 for theophylline is 10 mg/L. The Emax is equivalent to 63% of normal FEV1. A further increase in the plasma theophylline concentration will not yield an improvement in the FEV1 beyond Emax. Either drug saturation of the receptors or other limiting factors prevent further improvement in the pharmacologic response.

The Emax model describes two key features of the pharmacologic response: (1) the model mimics the hyperbolic shape of the pharmacologic response–drug concentration curve, and (2) a maximum pharmacologic response (Emax) may be induced by a certain drug concentration, beyond which no further increase in pharmacologic response is obtained (see Fig. 19-17). The drug concentration that produces a 50% maximum pharmacologic response (EC50) is useful as a guide for achieving drug concentration that lies within the therapeutic range.

In many cases, the measured pharmacologic effect has some baseline value when drug is absent (eg, blood pressure, heart rate, respiration rate). E0 is the measured pharmacologic effect (baseline activity) at zero drug concentration in the body. The measurement for E0 may be variable due to intra- and intersubject differences. Using E0 as a baseline constant-effect term, Equation 19.13 may be modified as follows:

The effect may be expressed as a fraction or a percentage of maximum effect Emax, and so Equation 19.14 becomes 19.14a:

where, E0′ = E0/Emax.

Sigmoid Emax Model

The sigmoidEmaxmodel describes the pharmacologic response–drug concentration curve for many drugs that appear to be S shaped (ie, sigmoidal) rather than hyperbolic as described by the simpler Emax model. The model was first used by Hill (1910) to describe the association of oxygen with hemoglobin, in which the association with one oxygen molecule influences the association of the hemoglobin with the next oxygen molecule. The equation for the sigmoid Emax model is an extension of the Emax model:

where n is an exponent describing the number of drug molecules that combine with each receptor molecule. When n is equal to unity (n = 1), the sigmoidal Emax model reduces to the Emax model. A value of n > 1 influences the slope of the curve and the model fit.

The sigmoidal Emax model has been used to describe the effect of tocainamide on the suppression of ventricular extrasystoles (Winkle et al, 1976). As shown in Fig. 19-19, the very steep slope of the tocainamide concentration–response curve required that n = 20 in order to fit the model. Although this model was developed empirically, the mathematical equation describing the model is similar to the one elaborated by Wagner (1968) and discussed earlier in this chapter.

In the sigmoid Emax model, the slope is influenced by the number of drug molecules bound to the receptor. Moreover, a very large n value may indicate allosteric or cooperative effects in the interaction of the drug molecules with the receptor.

Pharmacokinetic Pharmacodynamic Models with an Effect Compartment

Many pharmacokinetic models describe the time course for drug and metabolite concentrations in the body. Using either the sigmoid Emax or one of the other pharmacodynamic models described earlier, the pharmacologic response may be obtained and modeled at various time periods. This simple approach has worked for some neuromuscular blockers and anesthetic agents, whose activities are related to plasma drug concentrations.

For some drugs, the time course for the pharmacologic response may not directly parallel the time course of the plasma drug concentration. The maximum pharmacologic response produced by the drug may be observed before or after the plasma drug concentration has peaked. Moreover, other drugs may produce a delayed pharmacologic response unrelated to the plasma drug concentration.

A pharmacokinetic/pharmacodynamic model with an effect compartment is used to describe the pharmacokinetics of the drug in the plasma and the time course of a pharmacologic effect of a drug in the site of action. To account for the pharmacodynamics of an indirect or delayed drug response, a hypothetical effect compartment has been postulated (Fig. 19-20). This effect compartment is not part of the pharmacokinetic model but is a hypothetical pharmacodynamic compartment that links to the plasma compartment containing drug. Drug transfers from the plasma compartment to the effect compartment, but no significant amount of drug moves from the effect compartment to the plasma compartment. Only free drug will diffuse into the effect compartment, and the transfer rate constants are usually first order. The pharmacologic response is determined from the rate constant, ke0, and the drug concentration in the effect compartment (see Fig. 19-20).

The amount of drug in the hypothetical effect compartment after a bolus IV dose may be obtained by writing a differential equation describing the rate of change in drug amounts in each compartment:

where De is the amount of drug in the effect compartment, D1 is the amount of drug in the central compartment, k1e is the transfer rate constant for drug movement from the central compartment into the effect compartment, and ke0 is the transfer rate constant out of the effect compartment.

Integrating Equation 19.16 yields the amount of drug in the effect compartment De:

Dividing Equation 19.17 by Ve, the volume of the effect compartment, yields the concentration Ce of the effect compartment:

where D0 is the dose, Ve is the volume of the effect compartment, and k is the elimination rate constant from the central compartment. Equation 19.18 is not very useful because the parameters Ve and k1e are both unknown and cannot be obtained from plasma drug concentration data. Several assumptions were made to simplify this equation.

The pharmacodynamic model assumes that even though an effect compartment is present in addition to the plasma compartment, this hypothetical effect compartment takes up only a negligible amount of the drug dose, so that plasma drug level still follows a one-compartment equation. After an IV bolus dose, the rate of drug entering and leaving the effect compartment is controlled by the incoming rate constant k1e and the elimination rate constant ke0. (There is no diffusion of drug from the effect compartment into the plasma compartment.) At steady state, both the input and output rates from the effect compartment are equal,

Rearranging,

Dividing by VD yields the steady-state plasma drug concentration C1:

Substituting for De into Equation 19.21 yields

Cancelling the common term k1e,

At steady state, C1 is unaffected by k1e and is controlled only by the elimination constant k and ke0. C1 is called Cpss, or steady-state drug concentration, and has been used successfully to relate the pharmacodynamics of many drugs, including some with delayed equilibration between the plasma and the effect compartment. Thus, k and ke0 jointly determine the pharmacodynamic profile of a drug. In fitting the pharmacokinetic–pharmacodynamic model, the IV bolus equation is fitted to the plasma drug concentration–time data to obtain k and VD, while Cpss, or C1 from Equation 19.24, is used to substitute into the concentration in Equation 19.15 to fit the pharmacologic response.

Many drug examples have been described by this type of pharmacokinetic–pharmacodynamic model. The key feature of this model is its dynamic flexibility and adaptability to pharmacokinetic models that account for drug distribution and pharmacologic response. The aggregate effects of drug elimination, binding, partitioning, and distribution in the body are accommodated by the model. The basic assumptions are practical and pragmatic, although some critics of the model (Colburn, 1987) believe the hypothetical effect compartment may oversimplify more complex drug–receptor events. On the positive side, the model represents elegantly an in vivo pharmacologic event relating to the plasma drug concentrations that a clinician can monitor and adjust.

Until more information is known about the effect compartment, a pharmacokinetic–pharmacodynamic model is proposed to describe these kinetic processes combining some of the variables. A good fit of the data to the model is useful but does not necessarily describe the actual pharmacodynamic process. The process of model development evolves until a better model replaces an inadequate one. Several examples of drugs incorporating the effect compartment concept cited in the next section support the versatility of this model. The model accommodates some difficult drug response–concentration profiles, such as the puzzling hysteresis profile of some drug responses (eg, responses to cocaine and ajmaline).

Pharmacodynamic Models Using an Effect Compartment

The antiarrhythmic drug ajmaline slows the heart rate by delaying the depolarization of the heart muscle in the atrium and the ventricle. The pharmacologic effect of the drug is observed in the ECG by measuring the prolongation of the PQ and QRS interval after an IV infusion of ajmaline. A two-compartment model with binding described the pharmacokinetics of the drug and a pharmacodynamic model with an effect compartment was linked to the central compartment in which free drug may diffuse into the effect compartment. The effect compartment was necessary because the plasma ajmaline concentration did not correlate well with changes in recorded ECG events. When the effect-compartment drug concentration was used instead, drug activity was well described by the model (Figs. 19-21 and 19-22).

Hysteresis of Pharmacologic Response

Many pharmacologic responses are complex and do not show a direct relationship between pharmacologic effect and plasma drug concentration. Some drugs have a plasma drug concentration–pharmacologic response that resembles a hysteresis loop (Fig. 19-23). For these drugs, an identical plasma concentration can result in significantly different pharmacologic responses, depending on whether the plasma drug concentration is on the ascending or descending phase of the loop. The time-dependent nature of a pharmacologic response may be due to tolerance, induced metabolite deactivation, reduced response, or translocation of receptors at the site of action. This type of time-dependent pharmacologic response is characterized by a clockwise profile when pharmacologic response is plotted versus plasma drug concentrations over time (see Fig. 19-23).

For example, fentanyl (a lipid-soluble, opioid anesthetic) and alfentanil (a closely related drug) display clockwise hysteresis, apparently due to rapid lipid partition. β-Adrenoreceptors, such as isoproterenol, apparently have no direct relationship between response and plasma drug concentration and show hysteresis features. The diminished pharmacologic response was speculated to be a result of cellular response and physiologic adaptation to intense stimulation of the drug. A decrease in the number of receptors as well as translocation of receptors was proposed as the explanation for the observation. The euphoria produced by cocaine also displayed a clockwise profile when responses were plotted versus plasma cocaine concentration (Fig. 19-24).

A second type of pharmacologic response shows a counterclockwise hysteresis profile (Fig. 19-25). The pharmacologic response increases with time as the pharmacologic response is plotted versus plasma drug concentrations. An example of a counterclockwise hysteresis loop is the antiarrhythmic drug ajmaline. When the QRS interval changes in dogs were plotted versus plasma ajmaline concentration in each dog, an interesting counterclockwise hysteresis loop was seen (see Fig. 19-21). Yasuhara and co-workers (1987) developed a pharmacodynamic model to analyze the molecular events between drug concentration and change in ECG parameters such as QRS. A relationship was established between pharmacologic response and drug concentration in the effect-compartment drug level (see Fig. 19-22). The hysteresis profile (see Fig. 19-21) is the result of the drug being highly bound to the plasma protein (α1-acid glycoprotein), and of a slow initial diffusion of drug into the effect compartment.

Counterclockwise hysteresis curves may also result when the measured pharmacodynamic response is not the primary effect of the drug, ie, there is an indirect effect. For example, warfarin inhibits hepatic synthesis of clotting factors II, VII, IX, and X, but prothrombin time is measured as a surrogate for warfarin activity and clotting factor concentration.

To predict the time course of drug response using a pharmacodynamic model, a mathematical expression is developed to describe the drug concentration–time profile of the drug at the receptor site. This equation is then used to relate drug concentrations to the time course and intensity of the pharmacologic response. Most pharmacodynamic models assume that pharmacologic action is due to a drug–receptor interaction, and the magnitude of the response is related quantitatively to the drug concentration in the receptor compartment. In the simplest case, the drug receptor lies in the plasma compartment and pharmacologic response is established through a one-compartment model with drug response proportional to log drug concentration (Equation 19.1). A more complicated model involving a receptor compartment that lies outside the central compartment was proposed by Sheiner and associates (1979). This model locates the receptor in an effect compartment in which a drug equilibrates from the central compartment by a first-order rate constant k1e. There is no back diffusion of drug away from the effect compartment, thereby simplifying the complexity of the equations. This model was applied successfully to monitor the pharmacologic effects of the drug trimazosin (Meredith et al, 1983).

The pharmacokinetics of trimazosin are described as a two-compartment open model with conversion to a metabolite by a first-order rate constant k1m. The pharmacokinetics of the metabolite are described by a one-compartment model with a first-order elimination constant km0. The drug effect may be described by two pharmacodynamic models, either model A or B. Model A assumes that the drug effect in the effect compartment is produced by the drug only. Model B assumes that both the drug and a metabolite produce drug effect (Fig. 19-26).

The following equation describes the pharmacokinetics and pharmacodynamics of the drug:

where Cp is the concentration of the drug in the central compartment.

where Cm is the concentration of the metabolite in the body, Vm is the volume of distribution of the metabolite, V1 is the volume of the central compartment of the body, k1m is the first-order constant for converting drug to metabolite, km0 is the elimination rate constant of the metabolite, A and B are two-compartment model coefficients for the drug (see Chapter 4), and k10 is the elimination rate constant of the drug.

The drug concentration in the effect compartment is calculated by assuming that at equilibrium the concentration of the drug in the effect compartment and the central compartment are equal,

where Ve is volume of the effect compartment and keq is the elimination rate constant of the drug from the effect compartment. Therefore, the drug concentration in the effect compartment C(e, d) is calculated as

The effect due to drug is assumed to be linear,

where Md is the sensitivityslope to the drug (ie, the effect per unit of drug concentration in the effect compartment). The parameters Md, i, and keq are determined by least-squares fitting of the data. For the metabolite, the concentration of metabolite in the effect compartment is C(e, m).

The concentration of the metabolite in the effect compartment is in turn related to drug effect as for the parent drug. The total effect produced is

The five parameters Md, Mm, i, keq, keqm may be estimated from Equation 19.31 by fitting the data to an appropriate model. Figure 19-27 shows the observed decline in systolic blood pressure compared with the theoretical decline in blood pressure predicted by the model. An excellent fit of the data was obtained by assuming that both drug and metabolite are active. This example illustrates that, for a dose of a drug, the drug concentration in the effect compartment and others may be described by a mathematical model. These equations were further developed to describe the time course of a pharmacologic event. In this case, Meredith and associates (1983) demonstrated that both the drug and the metabolite formed in the body may affect the time course of the pharmacologic action of the drug in the body.

Simulation of In Vitro Pharmacodynamic Effect Involving Hysteresis

An in vitro model simulation of the sum of pharmacologic effect contributed by a drug and its active metabolite may explain the observation of the hysteresis response curve in vivo. Gupta et al (1993) discussed the factors that affect the shape of the response curve. In the simplest case, pharmacokinetic equations are developed to calculate Cp, the drug concentration, and Cm, the metabolite concentration. To estimate the pharmacologic effect due to both the drug and active metabolite, the potency of the drug is defined as P, the potency of the metabolite is Pm, and the sum of the pharmacologic effect is as shown below. (In their first simulation, Gupta et al assumed that the effect is linearly related to drug and metabolite concentrations.)

The shape of hysteresis simulated is very dependent on Pm and km0, the rate constant of metabolite elimination. If km0 is given a high, medium, or low value, the effect on the shape of the hysteresis loop is changed dramatically, as shown in Fig. 19-28. A temporal effect causes a counterclockwise loop. In the case of a metabolite that acts as an antagonist, the hysteresis loop is clockwise. The more elaborate features of an Emax model were simulated by Gupta et al (1993) in their paper.

Lorazepam Pharmacodynamics—Example of an In Vivo Hysteresis Loop

Many drugs that act on the central nervous systems (CNS) have a lag time before the tissues and the plasma are equilibrated with drug. The pharmacokinetics of lorazepam after oral absorption were fitted to a two-compartment model with lag time. Lorazepam was studied because the drug accounts for all the activity, such that the counterclockwise response profile may be attributed to equilibration rather than to metabolism (Gupta et al, 1990).

The description of the plasma drug concentrations, Cp, is obtained by conventional pharmacokinetic equations, whereas the pharmacodynamic effect, E, is described by a sigmoid Emax model similar to Equation 19.15, except that the baseline effect is also included. Gupta et al (1990) monitored three pharmacodynamic effects due to lorazepam. The monitored pharmacodynamic effects were mental impairment processes evaluated by the cognitive and psychomotor performance of the subjects, including (A) subcritical tracking, (B) sway open (a measurement of gross body movements), and (C) digital symbol substitution. When the time course of each effect was plotted versus plasma drug concentration, a counterclockwise loop was observed (Fig. 19-29). When the same pharmacodynamic responses were plotted versus lorazepam concentration in the effect compartment accounting for the equilibration lag, a classical sigmoid relation was observed (Fig. 19-30). The observations showed that the temporal response of many drugs may be the result of pharmacodynamic and distributional factors interacting with each other. Thus, a model with an effect compartment can more fully help understand the time course of the drug response.

Simulation of Antiplatelet Effect of a Mechanism-Based Pharmacodynamic Model for Aspirin

Low-dose aspirin, ASA (50–325 mg), has been shown to reduce the risk of myocardial infarction and stroke and is indicated for its cardioprotective effect. ASA acts by irreversibly acetylating a serine residue at position 529 in platelet cyclo-oxygenase-1 (COX-1), resulting in reduced production of proaggregatory thromboxane A2 (TXA2). Once COX-1 has been acetylated by aspirin, the access of substrate arachidonic acid to the catalytic site is impeded for the remainder of the platelet lifespan (8–10 days). Although ASA has an effect on COX-2, the cardioprotective effect is adequately explained by its inhibition of platelet COX-1 by low doses and is sufficient to explain the cardio-protective effect of aspirin observed in clinical trials (Altman et al, 2004).

Hong et al (2008) developed a mechanism-based pharmacodynamic model that characterizes the antiplatelet effects of aspirin and ibuprofen alone and in combination (see Fig. 19-31). Ten healthy volunteers were enrolled in a single-blinded, randomized, three-way crossover study. Treatments consisted of single doses of either oral aspirin (325 mg) or ibuprofen (400 mg) and concomitant administration of aspirin (325 mg) and ibuprofen (400 mg). Ex vivo whole blood platelet aggregation induced by collagen (1 μg/mL) or arachidonic acid (0.5 mmol/L) was measured by impedance aggregometry. Development and population parameterestimation were performed using nonlinear mixed-effects modeling implemented in NONMEM. The PK–PD model shows inhibition of platelet aggregation was achieved following aspirin treatment (˜77% inhibition within 2 hours), and return to baseline values occurred within 72 to 96 hours after dosing. In contrast, treatment with ibuprofen produced transient inhibition of platelet aggregation, with complete recovery observed in 6 to 8 hours. The PK–PD simulation predicts a significant inhibition of aspirin antiplatelet effects in the presence of a typical ibuprofen dosing regimen.

Model simulations are used to predict the relationship between drug exposure and pharmacologic response in situations where real experimental data are sparse or absent. There are many different types of models for the analysis of exposure-response data: (1) descriptive PD models (Emax model for exposure-response relationships), (2) empirical models that link a PK model (dose-concentration relationship), and (3) PD model (concentration-response relationship).

Descriptive or empirical model-based analysis does not necessarily establish causality or provide a mechanistic understanding of a drug's effect. Surrogateendpoints are a subset of biomarkers. A surrogate endpoint is a laboratory measurement or physical sign used in therapeutic trials as a substitute for a clinically meaningful endpoint that is expected to predict the effect of the therapy (Temple, 1999). Adequate and well-controlled clinical studies that investigate several fixed doses and/or measure systemic exposure levels, when analyzed using scientifically reasonable causal models, can predict exposure-response relationships for safety and/or efficacy and provide plausible hypotheses about the effects of alternative doses and dosage regimens not actually tested (FDA guidance: Exposure-Response Relationships 2003). The broad term exposure refers to dose (drug input to the body) and various measures of acute or integrated drug concentrations in plasma and other biological fluid (eg, Cmax, Cmin, Css, AUC). Response refers to a direct measure of the pharmacologic effect of the drug. Response includes a broad range of endpoints or biomarkers ranging from the clinically remote biomarkers (eg, receptor occupancy) to a presumed mechanistic effect (eg, ACE inhibition), to a potential or accepted surrogate (eg, effects on blood pressure, lipids, or cardiac output), and to the full range of short-term or long-term clinical effects related to either efficacy or safety.

Both agonist and antagonist drug effect can be quantitatively simulated by PK–PD models. The most common models are Emax models mechanistically based on drug receptor theory. Although most drug responses are complex, pharmacologic response versus log dose type of plots have been shown to follow sigmoid type of curve (S-curve) with maximum response peaking when all receptors become saturated. In vitro screening preparations are useful to study EC50, potency, and mechanism of a drug. However, pharmacologic response in a patient is generally far more complicated. Physiologically based PD models must consider how the drug is delivered to the active site and the effect of various drug disposition processes, as well as plasma and tissue drug binding. In addition, pharmacogenomics of the drug and disease processes must be considered in the model. Appropriately developed PK–PD models may be applied to predict onset, intensity, and duration of action of a drug. Toxicokinetics may also be applied to explain the side effects or drug-drug interactions.

The progress of a disease or its response to a therapeutic agent is often accompanied by biologic changes (markers or biomarkers) that are observable and/or measurable. Biomarkers (BM) may be selected and validated to monitor the course of drug response in the body. BMs should be mechanistically based and fulfill a number of clinically relevant criteria in order to be useful as potential clinical endpoints. BM together with PK–PD could be a very useful tool in expediting drug development, and many reviews and discussions are available about this application.

1. On the basis of the graph in Fig. 19-32, answer “true” or “false” to statements (a) through (e) and state the reason for each answer.

   1. The plasma drug concentration is more related to the pharmacodynamic effect of the drug compared to the dose of the drug.

   2. The pharmacologic response is directly proportional to the log plasma drug concentration.

   3. The volume of distribution is not changed by uremia.

   4. The drug is exclusively eliminated by hepatic biotransformation.

   5. The receptor sensitivity is unchanged in the uremic patient.

   Figure 19-32Graphic Jump Location

   

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   Graph of pharmacologic response E as a function of time for the same drug in patients with normal (A) and uremic (B) kidney function, respectively.

2. What do clavulanate, sulbactam, and tazobactam have in common? Why are they used together with antibiotics?

3. Explain why subsequent equal doses of a drug do not produce the same pharmacodynamic effect as the first dose of a drug.

   1. Provide an explanation based on pharmacokinetic considerations.

   2. Provide an explanation based on pharmacodynamic considerations.

4. How are the parametersAUC and teff used in pharmacodynamic models?

5. What class of drug tends to have a lag time between the plasma and the effect compartment?

6. Name an example of a pharmacodynamic response that does not follow a drug dose–response profile?

7. When an antibiotic concentration falls below the MIC, there is a short time period in which bacteria fail to regrow because of postantibiotic effect (PAE). This time period is referred to as PAT. What is PAT?

8. What is AUIC with regard to an antibiotic?

9. What is the difference between IC50 and EC50? Are the values reproducible from one lab to another?

   In functional studies, the antagonist IC50 is most useful if the concentration of the agonist is below maximal. Higher concentrations of the agonist will increase the IC50 of the competitive antagonist well above its equilibrium dissociation constant. Even with low agonist concentrations, the IC50 from functional studies, like an agonist EC50 or maximal response, is dependent on the conditions of the experiment (tissue, receptor expression, type of measurement, etc). True or false?

10. Ki refers to the equilibrium dissociation constant of a ligand determined in inhibition studies. The Ki for a given ligand is typically determined in a competitive radioligand binding study by measuring the inhibition of the binding of a reference radioligand by the inhibiting ligand of under equilibrium conditions, Why?

11. What is the dissociation constant K in the following interaction between a drug Ligand L

    

    and a drug receptor R

    

    where K is expressed as k−1/k+1 and PLR is the proportion of receptor occupied by L.

    How many binding sites are assumed in the above model?

12. According to Katzung et al, 2009, propranolol is an example of a competitive antagonist (ie, β-adrenoceptor antagonist). The degree of inhibition produced depends on its concentration in the body. Some patients receiving a fixed dose of propranolol exhibit differences in inhibitory effects on physiologic responses to norepinephrine and epinephrine (endogenous adrenergic receptor agonists, based on drug receptor theory), and the dose of propranolol must be adjusted accordingly. Based on your understanding of drug receptor theory and propranolol PK (refer to Appendix E), which of the following may contribute to propranolol response differences?

    (a) Clearance, (b) protein binding, and (c) bioavailability of the drug

    

    Applying Equation 19.12b above, for a one- binding site reaction, substituting b = 1, and the variable z in the above equation may be seen to be essentially [L], it can be seen that the Hill equation is applicable to this simple case of [L] binding to a receptor having one binding site.

13. Which one of the following would you select as a biomarker for a Type II diabetic patient? State the reasons that support your selection.

    1. Blood sugar level

    2. Blood insulin level

    3. HbA1C