General Principles of Pharmacokinetics and Pharmacodynamics

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169 General Principles of Pharmacokinetics and Pharmacodynamics

Critically ill patients admitted to intensive care units (ICUs) suffer from a variety of physiologic insults that accompany their severe illness. These insults, combined with the rapidly changing physiologic status of the patient, can make appropriate drug dosing a challenging problem for the clinician. An understanding of the pharmacokinetic implications of these physiologic changes and their subsequent effect on pharmacodynamics is required to properly treat critically ill patients. This chapter reviews the basic principles of pharmacokinetics and pharmacodynamics with an emphasis on how they might be affected by critical illness.

Pharmacokinetics and pharmacodynamics describe, respectively, the amount of drug in the body at a given time and the pharmacologic effects caused by the drug.¹ Pharmacokinetics describes the movement of a drug into, within, and out of the body over time, whereas pharmacodynamics explains the effects the drug has on the body that result in a clinical response. A general understanding of pharmacokinetic parameters such as clearance, volume of distribution, half-life, steady state, and absorption, along with pharmacodynamic principles such as receptor theory, potency, affinity, tolerance, and minimum effective concentration greatly enhances the clinician’s ability to make informed choices in the treatment of the critically ill patient.

General Principles of Pharmacokinetics

Clearance, volume of distribution, half-life, and bioavailability are four pharmacokinetic parameters that allow the clinician to better estimate dosing requirements. If the concentration of a drug in an easily assessable sampled fluid (e.g., plasma, urine, saliva) correlates well with the pharmacologic response (therapeutic or toxic) to the drug, then the application of pharmacokinetics in dosing is likely to be beneficial.² Usually the concentration of a drug cannot be measured at the exact site of action (e.g., a receptor on the cell surface), so it is necessary that there be a predictable relationship between the concentration that is measurable and the concentration at the site of the effect.^3,⁴ These concentrations do not have to be equal, but they should reflect a similar direction and magnitude of change over time (Figure 169-1).

Figure 169-1 For concentration monitoring to be useful, there must be a strong relationship between concentration of the drug measured in an easily accessible fluid and concentration at the effect site. Concentration at the effect site may be less, more, or equal to concentration in the sampled fluid.

Measurement of the relationship between drug concentration and therapeutic or toxic response in a large number of patients allows development of a therapeutic range or target concentration for that drug (Figure 169-2).⁵^–¹⁰ Table 169-1 lists a number of drugs commonly used in the ICU for which therapeutic ranges have been established and for which therapeutic drug monitoring is often recommended.^11,¹² Critically ill patients have a multitude of host factors (e.g., hemodynamic status, decreased organ function, nutritional status, concurrent disease states) that increase the likelihood that individualized drug dosing based on individualized pharmacokinetic assessment will be beneficial (Figure 169-3).¹³^–¹⁶ There can be gender-related differences in both pharmacokinetic and pharmacodynamic responses.¹⁷^–¹⁹ Individual chapters in this text are devoted to many of these agents and their adjustments for dosage in patients with renal or hepatic failure.

Figure 169-2 Therapeutic range represents the concentration at which a desired effect is likely to occur in most patients and an adverse or toxic effect is rare. If such a range cannot be established, concentration monitoring for the drug is not likely to be of benefit. Therapeutic range is often established by dose-ranging studies during phase 2 drug development and confirmed during the phase 3 trial.

TABLE 169-1 Therapeutic Ranges of Drugs Commonly Used in Critical Care

Drug	Therapeutic Range
Amikacin	Trough < 5 µg/mL
Peak < 30 µg/mL
Cyclosporine	Whole blood, 150 ng/mL
Digoxin	0.50-2.0 ng/mL
Gentamicin*	Trough < 2 µg/mL
Peak < 10 µg/mL
Lidocaine	1.5-5 µg/mL
Phenytoin	10-20 µg/mL
Quinidine	2-5 µg/mL
Theophylline	10-20 µg/mL
Tobramycin*	Trough < 2 µg/mL
Peak < 10 µg/mL
Vancomycin^†	Trough < 5 µg/mL
Peak < 30 µg/mL

* Once daily aminoglycoside dosing may result in different therapeutic ranges.

^† Vancomycin concentrations are currently focusing on higher peak concentrations, and practice varies considerably between sites.

Figure 169-3 Example of interacting factors that determine the effect seen after administration of a single drug dose in an individual patient in the intensive care unit. A patient experiencing shock has decreased drug clearance by the liver and kidney; slowed absorption of oral, intramuscular, or topical medications; and a highly variable volume of distribution based on fluid status.

Pharmacokinetic Models

The pharmacokinetic concepts of clearance, volume of distribution, half-life, and bioavailability are based on physiologic principles.²⁰ The physiologic processes governing these concepts are enormously complex, and many simplifying assumptions must be made before the mathematics describing drug concentrations become tractable. Although sophisticated computer modeling approaches are available in research settings, most of the clinically useful pharmacokinetic equations are based on one- or two-compartment models (Figure 169-4).²¹

Figure 169-4 In the simplest pharmacokinetic model, the body is treated as a single compartment into which drug is delivered and eliminated. The resulting concentration in the compartment defines the apparent volume of distribution. Most drugs follow the more complicated two-compartment model, which assumes a distribution phase between the central or plasma compartment and the tissue. See text for explanation of terms.

The simplest model and the most basic equations describe the one-compartment model. When the drug enters the compartment, it is assumed to be instantaneously and completely mixed in a given volume of distribution (V) resulting in a uniform concentration throughout the compartment. The parameter K is the first-order rate constant that reflects the usual situation of elimination being a first-order linear process. The drug is assumed to enter the compartment instantaneously in the case of an intravenous bolus dose. If the dosage is administered through oral or intramuscular routes, entry into the compartment is assumed to occur at a rate defined by a first-order absorption rate constant (Ka). Entry into the compartment is assumed to occur at a rate described by a zero-order rate constant (Ro) if the drug is administered by constant intravenous infusion. Bioavailability (F) is defined as the fraction of the administered dose that reaches the systemic circulation.

Clearance (CL) is a primary parameter that can be physiologically associated with a particular organ in the body such as the liver or kidney. Clearance can be calculated according to the equation CL = K × V, leading to the impression that CL is a function of the parameters K and V. However, this arrangement of the equation is not correct from a physiologic point of view. CL and V are both primary parameters, and K is a secondary parameter. The first-order rate is determined by changes in either CL or V, and the equation is correctly written: K = CL/V.

Half-life (t_1/2) is a useful measure of how quickly a drug is eliminated from the body, and it is related to the first-order elimination rate constant:

Specifically, t_1/2 defines the length of time it takes for the drug concentration to decrease by one-half. In a linear pharmacokinetic system with first-order elimination, the t_1/2 is a constant, and it takes the same amount of time for the concentration to fall from 100 to 50 arbitrary units as it does to decline from 50 to 25 arbitrary units (Figure 169-5).

Figure 169-5 Log concentration-time curve for a one-compartment model after intravenous administration, illustrating volume of distribution, elimination rate constant (K), and half-life (t_1/2). C1, C2, and C3 are measured drug concentrations.

The single-compartment model allows concentration at any point in time to be calculated using the following equation:

where Δt is the time elapsed between the measurement of two concentrations, C1 and C2. It is the properties of this equation that give rise to the familiar exponentially decreasing concentration-time curve, which becomes linear when plotted on semilog coordinates.

The human body is not a single well-stirred compartment, and it is amazing that such a simple mathematical model can be so useful in the clinical setting. If the body is conceptualized as consisting of individual tissues and organs, the same mathematical treatment can be applied. The concept of the volume of distribution has to be somewhat modified to recognize not only the physical size of the organ or tissue, but also the fact that drugs accumulate to differing degrees in different tissue spaces.^22,²³ For example, lipophilic drugs have a high affinity for adipose tissue, and this property is reflected by a large partition coefficient (R). The time constant associated with each tissue is a function of the rate of blood flow to that tissue (Q), the physical volume of the tissue (V_T), and the partition coefficient. The time constant determines the rate at which equilibrium is reached. As a result, it is possible to construct a set of exponential equations with a time constant unique to each tissue:

There is a branch of pharmacokinetics known as physiologically based pharmacokinetic modeling that uses blood flows, organ volumes, and partition coefficients to characterize concentration-time profiles.²⁴^–²⁷ According to this approach for pharmacokinetic modeling, each tissue space ultimately contributes to the venous pool (Figure 169-6). However, the overall shape of the concentration-time profile in the venous blood is controlled not by the number of tissue spaces or their effective volumes, but by their time constants. Tissues with similar time constants, Q/(V × R), produce similar drug profiles in their venous outflows and appear as a single exponent in pooled venous blood. Practically speaking, many tissues and organs reach equilibrium over similar time frames, and often no more than two distinct time constants are observed. Therefore, this situation can be described adequately by a two-compartment model, characterized by a rapidly distributing central compartment and a more slowly equilibrating peripheral compartment (Figure 169-7). The equation describing the concentration-time profile for the two-compartment model is:

Figure 169-6 Physiologically based models allow individual characterization of drug distribution or clearance for each organ or tissue and also describe the mixed volume effects seen when sampling from blood.

Figure 169-7 Log concentration-time curve for a two-compartment model after intravenous push administration, illustrating a distribution period (α) and postdistribution period (β). Concentrations at C1 and C2 are reflective of both distribution and elimination processes, whereas concentrations at C3 and C4 are primarily affected by elimination processes (clearance).

The distinguishing feature of this biexponential equation is that when it is plotted on semilog coordinates, the concentrations are the sum of two distinct straight lines. Hence, there are two half-lives. One is known as the terminal or β half-life, and the other is the rapid distribution or α half-life. Once the rapid distribution exponential becomes negligible in the equation, all that remains is the slower exponential term, and the concentration-time profile resembles that for a single-compartment drug. Consequently, the equation:

in which β replaces K, can still be used to predict concentrations, as long as both C1 and C2 are in the postdistributive phase. This sum-of-exponentials approach can be extended to three-compartment or even more complex models, but it is difficult to obtain all the concentrations needed to characterize each exponent.

Clearance

Clearance (CL) is a primary pharmacokinetic parameter that measures the ability of the body to eliminate a drug.^28,²⁹ It is often stated that clearance is the volume of blood (plasma) that is completely cleared of drug per unit time. Although this is one way to define clearance, it does not capture the relationship between drug clearance (mL/min) and the rate of drug elimination (mg/h). In pharmacokinetics, the general concept of clearance is defined as the rate of elimination relative to the concentration. In a first-order pharmacokinetic system, the rate of elimination is proportional to the drug concentration, and clearance is this proportionality constant:

Clearance is clinically useful because it can be related directly to the organ of elimination. We can talk about renal clearance, hepatic clearance, or biliary clearance, and the sum of each of the individual clearances is the total body clearance.^30,³¹ The immediate clinical consequence is the ability to adjust doses in response to changes in specific organ function. For example, a patient with developing renal failure is likely to require a reduction of the dose of a drug that is eliminated by the kidney, but not necessarily a reduction of the dose of a drug that is eliminated by the liver.³² If the clearance of a drug is known to be 50% renal and 50% hepatic, and renal function is decreased by 50%, it is necessary to reduce the dose by only 25% to maintain the same concentration.

The primary clinical utility of clearance is that it is the single pharmacokinetic parameter that determines overall drug exposure. The area under the curve (AUC) on a plot of drug concentration as a function of time is often taken as a measure of drug exposure, and it is determined from the dose and clearance (CL):

This relationship is also observed when the steady-state concentration is considered as the measure of drug exposure. During a continuous intravenous infusion, the steady-state concentration (Css) is solely a function of the infusion rate (Ro) and the clearance (CL):

Notice that Css is not a function of the volume of distribution. As counterintuitive as it may seem, doubling the volume of distribution will not result in a halving of Css. The important point to keep in mind is that the equation is predicting the concentration at steady state. During a constant infusion at steady state, the rapid doubling of the volume of distribution will only transiently decrease the concentration by half. If the infusion rate remains unchanged, the concentration will return to the same steady-state concentration, as long as clearance remains unchanged.

The same principle applies to intermittent intravenous or oral dosing as well as continuous infusion. With intermittent dosing, drug concentrations go up and come down during each dosing interval. The average concentration at steady state (Css, avg) is a time-averaged concentration (i.e., the mean of all concentrations during the dosing interval); as in the case of a constant infusion, it is a function of clearance and the dosing rate. In the case of oral administration, the dosing rate becomes slightly more complicated, in that it is a function of the dose administered (D), the dosing interval (τ), and bioavailability (F):

As before, overall drug exposure is not influenced by volume of distribution, but it does change in proportion to changes in clearance or the dosing rate, through changes in F, D, or τ.

Volume of Distribution

The volume of distribution (V) is another primary pharmacokinetic parameter that is useful in determining the change in drug concentration for a given dose.³³ After an intravenous bolus dose in a one-compartment pharmacokinetic model, the change in concentration (ΔC) between Cmax and the concentration immediately before the dose is administered is a function of the dose (D) and the volume of distribution (V):

This equation is useful for predicting both the concentration after a first bolus dose and the increase in concentration at any point in time after a bolus dose. If a concentration before administration of a bolus dose is known or can be estimated, the equation can be used to predict the increase in concentration after the dose is administered (see Figure 169-5). It is important to recognize that the calculated value of ΔC must be added onto the pre-dose concentration to estimate the Cmax after the bolus dose. This equation is also useful for estimating the dose needed to reach a given concentration. If it is known that the volume of distribution is 0.45 L/kg, and a Cmax of 10 mg/L is desired after the loading dose, the dose is estimated to be 10 mg/L × 0.45 L/kg = 4.5 mg/kg. It is not necessary to have a steady-state condition to use this equation, a fact that makes it very useful in critical care.

The value for volume of distribution does not necessarily coincide with any particular physiologic space. The veracity of this statement becomes readily apparent when one considers a drug such as digoxin which has a volume of distribution of approximately 440 L. Clearly, a volume of distribution of that magnitude cannot have a relationship to any physiologic space in a standard-sized human. For this reason, the term apparent volume of distribution is often used.

The concept of volume of distribution gets more confusing when more than one compartment is needed to describe the pharmacokinetics of a drug. Mathematically, the volume of distribution is a hypothetical volume that is needed to relate the amount of drug in the body to a measured concentration in a fluid (usually plasma). Unlike the one-compartment model, wherein all of the drug in the body is regarded as being in a single compartment until it is eliminated, drug also circulates through additional compartments in a multicompartment model. In this situation, the volume of distribution must increase as drug distributes to other compartments until pseudodistribution equilibrium among all compartments is reached. Technically, an infinite number of volumes of distribution are observed as this equilibration process occurs, but only three are commonly defined. The volume of distribution of the central compartment (Vc) is the volume of the usual sampling compartment; it is always the smallest volume term. Immediately after administration of an intravenous bolus, all added drug is in the central compartment, and Vc can be used to calculate a change in concentration.

The volume of distribution increases over time until a distribution equilibrium is reached among all compartments. This is the largest value for the volume of distribution. The fact that distribution equilibrium has occurred can be determined from a log-concentration versus time plot (see Figure 169-7). The curve becomes log-linear when the rate of drug entry into each peripheral compartment equals the rate of exit from each compartment. Because it is often calculated using the clearance and the β or terminal elimination half-life, this volume is often called V_β:

The third commonly used volume term is the steady-state volume of distribution (Vss). It is the sum of the volumes of all the compartments in the model. If a drug were infused to steady state, Vss would be the proportionality constant relating the steady-state concentration to the total amount of drug in the body. Practically speaking, Vss is not often used in individualizing drug dosing.

Half-Life

The half-life (t_1/2) is a pharmacokinetic parameter defined as the length of time it takes to reduce the drug concentration by half (see Figure 169-5).³³ The half-life is referred to as a secondary parameter because it is a function of the two primary parameters, clearance and volume of distribution:

A change in either clearance or volume of distribution results in a proportional change in half-life.

Because the half-life characterizes how rapidly concentration decreases over time, the primary clinical application for this parameter is for determining how often to dose a drug. Drugs with rapid half-lives have to be dosed more frequently than drugs with longer half-lives. The dosing of aminoglycoside antibiotics exemplifies this concept. The half-life for an aminoglycoside is relatively short in patients with good renal function (high clearance), and the drug may have to be dosed every 6 hours. In patients with poor renal function, the half-life is relatively longer, and dosing may be prolonged to 12- or 24-hour intervals to maintain appropriate peak and trough concentrations. In the critical care patient, the development of renal failure can significantly change aminoglycoside clearance, and the accompanying change in drug half-life will necessitate a change in dosing interval.

In a one-compartment system with constant clearance and volume of distribution, drug half-life also is constant. However, in a multicompartment model, the volume of distribution increases over time as drug equilibrates into tissue compartments until V_β is reached. According to the previous equation, the half-life also increases over time and eventually reaches a maximum at t_1/2β (see Figure 169-7).

In multicompartment models, there is usually one half-life of interest for each compartment. These half-lives are derived from the hybrid time constants associated with each compartment. In a two-compartment model, these two exponentials are typically called α and β and are arbitrarily termed the rapid and slow exponents, respectively. These time constants give rise to the rapid or distribution t_1/2α and the slower or terminal t_1/2β. One useful way to think about distribution half-lives is analogous to the standard way of thinking about any half-life. In the one-compartment model, it takes five half-lives for 97% of the drug to be eliminated from the body. The situation is similar for each exponent, but the interpretation is that it takes five distribution half-lives for that exponent to become negligible in the sum of exponentials equation. In other words, it takes five α half-lives before the rapid distribution phase is completed, and the remaining concentration-time profile reflects the elimination or β phase.

Most drugs have a rapid distribution phase that could be detected if concentrations were measured frequently enough. Aminoglycosides again are a good illustrative example of this concept, because they have a rapid, although not instantaneous, distribution phase (Figure 169-8). With a distribution phase half-life of 5 to 10 minutes, it would take approximately 25 to 50 minutes before the log-linear elimination phase could be observed. It is this distribution process that is the basis for the recommendation to wait approximately 1 hour after the end of an infusion before sampling blood to measure the aminoglycoside concentration. If a blood sample is obtained before this time, the drug still will be in the distribution phase, and the concentration measured will lead to underestimation of the drug half-life. In addition, slowly equilibrating compartments have been demonstrated when aminoglycoside concentrations are measured during washout.³⁴ Aminoglycosides are usually dosed frequently enough so that the slowly equilibrating compartment is not detected.