Patricia M. Dowling Successful antimicrobial therapy depends on both a measure of drug exposure (pharmacokinetics) and a measure of the potency of the drug against the infecting organism (pharmacodynamics). Pharmacokinetics is what the body does to the drug and pharmacodynamics is what the drug does to the pathogen. Efficacy relies on administering antimicrobial doses so that pathogens at the site of infection are killed or sufficiently suppressed that they can be eliminated by the host’s immune system. Pharmacokinetics (PK) is the mathematics of drug dosage determination, involving mathematical evaluation of the rates of drug absorption, distribution throughout the body, along with metabolism and ultimate excretion from the body. Pharmacodynamics is the effect of the antimicrobial drug on the pathogen relative to the drug concentration achieved at the site of infection (Chapter 5). Thus, effective antimicrobial therapy depends on pharmacokinetic/pharmacodynamic integration. Most pharmacokinetic information is derived from plasma drug concentrations. Antimicrobial action depends on drug concentration at the site of infection, but drug concentration measurement at the infection site is not always feasible. Instead, plasma (or serum) drug concentrations are measured and are assumed to represent drug concentrations in the target tissues. Most cells in the body are perfused with tissue fluids or plasma, and drug concentrations will usually reach an equilibrium between tissue fluids and the blood. For many antimicrobials, the pharmacodynamic action correlates well with the drug concentration in the blood. Therefore, measured plasma drug concentrations are generally assumed to represent drug concentrations at the infection sites in the tissues. The major exceptions to this relationship are the macrolide antimicrobials, whose lipid solubility results in lung tissue concentrations that bear little relationship to plasma concentrations. Mathematical models use equations to describe drug concentrations in the body as a function of time. In these models, the body is represented by a series of compartments that communicate reversibly with each other. A compartment is a tissue or group of tissues with similar blood flow and drug affinity. A drug is assumed to be uniformly distributed within a compartment. Drugs move dynamically in and out of compartments. Rate constants represent the entry and exit of drugs from each compartment. The typical compartment models used in pharmacokinetics include the: While the body can be divided into a vast number of compartments, models with more than three compartments are not physiologically and pharmacologically relevant. Most drugs used clinically are best described by two‐compartment models. Rate constants relate the observed rate of a kinetic process (drug absorption, drug elimination) to the variable (drug concentration) that controls the process. Some of the commonly determined rate constants in pharmacokinetic studies are: Reaction order is either zero order or first order; this refers to the way that drug concentration influences reaction rate. The amount of drug absorbed and/or eliminated changes at a constant time interval, regardless of the drug concentration. The rate of drug elimination is: where K0 is the zero order rate constant. For a drug with zero‐order elimination, a graph of drug concentration versus time on regular graph paper produces a straight line (Figure 4.1), with the equation: Figure 4.1 For a drug with zero‐order elimination, a graph of drug concentration versus time on regular graph paper produces a straight line. where C is the drug concentration at any time t, C0 is the drug concentration at time zero. For most drugs used therapeutically, zero‐order elimination occurs when elimination mechanisms become saturated. Renal tubular secretion and bile secretion of drugs are potentially saturable processes, so increasing drug doses show disproportional increases in systemic drug exposure. This phenomenon is seen with clinically used doses of rifampin. Drugs that normally do not undergo zero‐order elimination may show this if the patient has renal or hepatic insufficiency. Additionally, concurrently administered drugs may compete with each other for these elimination processes. Drugs that inhibit hepatic metabolism of other drugs may cause the inhibited drug to switch to zero‐order elimination (e.g., concurrent enrofloxacin inhibits the elimination of theophylline, resulting in toxicity). The amount of drug changes at a rate proportional to the amount of drug remaining in the body. The first order elimination rate is expressed as: where K is the first‐order rate constant and is expressed in units of time (min or hr). K defines the fraction of drug eliminated from the body per unit of time. C is the plasma drug concentration at any time (t). Although K remains constant, the rate (▴C/▴t) is always changing, because C is always decreasing. A graph of drug concentration versus time produces an exponential curve on regular graph paper, but produces a straight line on semilogarithmic graph paper (Figure 4.2), with the equation: where C is drug concentration at any time t, K is the first‐order rate constant in min or hr, and C0 is the drug concentration at time zero (the moment of injection). It is also written as C = C0e–kt. These concepts can be combined to mathematically describe the changes in the drug concentration in the body over time, as described below. This model considers the body to act as one homogeneous compartment (Figure 4.3). Drug concentration in one part of the body is assumed to be proportional to its concentration in any other part. When graphing concentration versus time data on semilog paper, the equation of the line is: or written in log base 10 terminology where C is the drug concentration at any time (t), C0 is the drug concentration at time zero (the moment of injection), K is the slope of the line (negative since the drug concentration is decreasing), and 2.3 is the constant needed to convert natural log to log base 10. This model describes many drugs administered by routes other than intravenous such as oral, subcutaneous, intramuscular, or intradermal routes. In an example of oxytetracycline (OTC) administered intramuscularly to horses (Figure 4.4), the plasma concentrations slowly rise as OTC is absorbed from the injection site into the plasma (central compartment). However, even as the drug is being absorbed, any drug in the central compartment is subject to the process of elimination. The curve peaks as these two processes reach equilibrium, and then declines as the amount of drug at the injection site depletes and elimination occurs only from the central compartment. This model describes the fate of most drugs administered IV to animals. The model assumes that the body acts as two compartments – the central compartment (blood and highly vascularized tissues) and a peripheral compartment (less vascularized tissues). K12 and K21 are the rate constants describing the movement of drug into and out of the peripheral compartment (Figure 4.5). Elimination only occurs from the central compartment (since the liver and kidneys are highly vascularized tissues). The concentration vs time line is not straight on semilog paper, unlike Figure 4.3, but rather it can be broken into two sections and described by the equation: where C is the concentration at any time (t), A is the y‐intercept of the first portion of the curve extrapolated to zero and α is the slope of the line, while B is the y‐intercept of the latter portion of the curve extrapolated to zero, and β is its slope. During the time described by the α portion of the line, drug is being distributed into the peripheral compartment as well as being eliminated from the central compartment. During the β portion, there is equilibrium of drug movement between the central and peripheral compartments, so the decline in plasma drug concentration is solely due to irreversible elimination from the central compartment. Figure 4.2 For a drug with first‐order elimination, a graph of drug concentration after IV admnistration versus time produces an exponential curve on regular graph paper, but produces a straight line on semilogarithmic graph paper. Figure 4.3 One‐compartment open model with IV injection and first‐order elimination shows a straight line of decreasing concentrations with a semilogarithmic scale. This model assumes that the entire body acts as one compartment with equal drug distribution within it. Figure 4.4 A one‐compartment open model with first‐order absorption and elimination describes oxytetracycline given IM to horses. Plasma concentrations rise with a rate constant of absorption (Ka) from the injection site into the central compartment (plasma and highly perfused tisses). During absorption, any drug in the central compartment is subject to elimination (K). The curve peaks as these two processes reach equilibrium, and then declines as the amount of drug at the injection site depletes and elimination occurs only from the central compartment. For some concentration versus time data, the line can be broken into three or more straight lines, and described mathematically with three or more exponential terms: With this equation, the first two terms describe distribution into peripheral compartments and the third describes the terminal elimination once equilibrium is reached. Theoretically, drug distribution in the body can be described by as many compartments as there are different tissues but for all practical purposes, more than three compartments models are not necessary (Figure 4.6). From a clinical standpoint, most drugs are best described by two‐compartment models. Drugs that are described by three‐compartment models usually have some tissue site where the drug is sequestered and slowly eliminated from the body. This is an important consideration for establishing withdrawal times in food‐producing animals. For example, aminoglycosides accumulate in renal tissues, which are routinely tested for antimicrobial residues. Figure 4.5 A two‐compartment open model with IV injection and first‐order elimination For this example of cefazolin injected intravenously into a dog, the initial portion of the line, defined mathematically as Ae−αt, is the distribution phase, where cefazolin is moving into tissues as well as being eliminated from the central compartment by glomerular filtration and renal excretion. K12 and K21 are the rate constants describing the movement of drug into and out of the peripheral compartment. The latter portion of the line, defined as Be−βt, is the elimination phase, where an equilibrium has been established between the central and peripheral compartments, so the line straightens out as drug is eliminated from the central compartment. After a drug is absorbed into the plasma, it is rapidly mixed and its molecules are distributed throughout the body by the systemic circulation. Drug molecules diffuse through networks of fine capillaries to tissue spaces filled with interstitial fluid. Interstitial fluid plus plasma makes up the extracellular fluid. Some drug molecules may further diffuse from the interstitial fluid across cell membranes into the cytoplasm (the intercellular fluid). The passage of drug molecules across a cell membrane is determined by the physical characteristics of the drug molecule, including ionization, lipid solubility, molecular size, and degree of protein binding. Cell membranes are composed of protein and a bilayer of phospholipid, making a “lipid–water–lipid sandwich.” Therefore, lipid‐soluble (nonpolar) drugs diffuse across cell membranes more easily than water‐soluble (polar) drugs. Small drug molecules will cross more easily than large drug molecules or drug molecules that are bound to proteins, such as albumin. Figure 4.6 A three‐compartment model with IV injection and first‐order elimination. For this example of gentamicin injected IV into a horse, the initial portion of the line, defined mathematically as Ae−αt, is the distribution phase, where gentamicin is moving into well‐perfused tissues as well as being eliminated from the central compartment by glomerular filtration and renal excretion. The middle portion of the line, defined as Be−βt, is the distribution phase into the deep compartment. Fpr gentamicin, this is the accumulation in the renal tubular epithelial cells. The terminal portion of the tine, defined by Ce−γt, is where an equilibrium has been established between the central and peripheral compartments, so the line straightens out as drug is eliminated from the central compartment. While difficult to demonstrate within the equation, there are rate constants describing the movement of drug into and out of the peripheral compartments and the central compartment with potential movement of drug between the peripheral compartments. Volume of distribution (Vd) (l/kg) is a mathematical term used to describe the apparent volume of the body in which a drug is dissolved (Toutain and Bousquet‐Melou, 2004d). Values for Vd are reported in most drug references and on drug labels. Although the Vd does not represent a true anatomical or physical volume, it does represent the dynamic drug distribution between the plasma and the tissues (Table 4.1). Three volumes of distribution (Vd) are typically reported in pharmacokinetic literature: the volume of the central compartment (Vdc), the volume of distribution calculated by the area method (Vdarea), and the steady‐state volume of distribution (Vdss). When not specified, it is usually a Vdss that is reported. Table 4.1 Relationship of volume of distribution to body water. It is useful to compare a drug’s Vd to the distribution of water in the body in order to get an idea of the drug’s distribution. Although the value of Vd generally predicts the extent of distribution of a drug, it does not confirm penetration of a drug to specific tissues. In general, the higher the value of Vd, the more likely it is that drug molecules will reach sequestered sites such as the brain and cerebrospinal fluid, prostate and other sex organs, eye, and mammary gland. Many studies still need to be done to confirm the concentrations of different drugs in specific tissues. For example, tulathromycin has extremely high Vd values in cattle and swine and achieves very high concentrations in lung tissue that persist for long periods of time, but concentrations in liver and kidneys are relatively low. As these are the target organs for residue testing, label slaughter withdrawal times are reasonable. Additionally, a high total tissue concentration does not guarantee an appropriate extracellular drug concentration for an antimicrobial, and total tissue concentration has no relevant intrinsic therapeutic meaning, especially when the pathogens are located in extracellular water of the target tissue. The value of the Vd is constant for any drug and only changes with physiological or pathological conditions that change the distribution of the drug. Clinically, any condition that changes extracellular fluid volume will dramatically affect the plasma concentrations of drugs with low Vd values. Drugs with high Vd values are not significantly affected by changes in body water status but may be affected by changes in body fat. Any condition that alters body water may require an adjustment in dose for drugs with low Vd values. A foal is 80% total body water, compared to an adult horse at 60% TBW, so that for any given dose of a drug with a low Vd like gentamicin or amikacin, because of dilution the foal will have lower plasma concentrations than the adult. Therefore, for a drug with a low Vd, a higher dose must be given to a foal (compared to the adult) to achieve the same effective plasma drug concentration. The degree of lipid solubility determines how readily a drug will cross biological membranes. Drugs are often classified as lipid soluble (or nonpolar) versus water soluble (or polar). Highly lipophilic drugs diffuse easily across almost all tissue membranes. Most of the antimicrobial drugs we use exist as weak acids or weak bases. Their lipid solubility depends a great deal (but not entirely) on their degree of ionization (charged state). An ionized drug such as penicillin G is hydrophilic and poorly lipid soluble. A nonionized drug such as erythromycin is lipophilic and can cross biological membranes. The degree of ionization for a weak acid or weak base depends on the pKa of the drug and the pH of the surrounding fluid. It is calculated from the Henderson–Hasselbach equation. When the pH is equal to the pKa of the drug, then the drug will be 50% ionized and 50% nonionized (log 1 = 0). Categorizing drugs according to their acid or base status shows the influence on the Vd, with a few exceptions (Table 4.2). Table 4.2 Influence of acid/base status on volume of distribution. Despite being weak bases, the aminoglycosides are very large, hydrophilic molecules and do not readily cross lipid membranes. Therefore, parenteral aminoglycosides do not achieve therapeutic concentrations in milk, prostatic or CSF fluids. Amphoteric drugs like the fluoroquinolones have both acidic and basic groups on their chemical structures. These drugs have a pH range where they are maximally nonionized. For example, enrofloxacin is most lipid soluble in the pH range of 6–8, so it is lipid soluble in most physiological situations. In acidic urine, however, significant ionization occurs, which reduces enrofloxacin antibacterial activity but this reduction in activity is offset by the extremely high concentration of enrofloxacin achieved in urine, so it is of no clinical importance. The effect of protein binding on the PK of drugs is frequently misunderstood and commonly incorrectly reported (Toutain and Bousquet‐Melou, 2002). The common blood proteins that bind drugs include albumin and globulins (Schmidt et al., 2010). Albumin is the major binding protein for acidic and neutral drugs, while globulin alpha‐1 acid glycoprotein (AGG) is the major binding protein for basic drugs. The degree of protein binding for an antimicrobial affects tissue penetration, elimination half‐life, and the Vd. First, only the unbound fraction of a drug in plasma penetrates into and equilibrates with the extravascular space where the majority of bacterial and fungal infections occur. Drugs that are highly protein bound typically have low values for Vd as they are primarily confined to the extracellular fluid compartment. Protein binding also affects drug clearance from the body as the protein‐bound drug serves as a “reservoir.” For antimicrobials eliminated by tubular secretion or hepatic metabolism, high protein binding is associated with slower drug elimination. Additionally, protein binding slows glomerular filtration, since only the free drug is filtered (Zeitlinger et al., 2011). For time‐dependent antimicrobials where keeping plasma concentrations above the bacterial MIC correlates with efficacy, a high degree of protein binding can be a clinical advantage. Compared to first‐generation cephalosporins (e.g., cephalexin, cefazolin), the third‐generation veterinary cephalosporins (e.g., ceftiofur, cefpodoxime, cefovecin) are highly protein bound, allowing administration with prolonged dosing intervals. The clinical significance of changes in the degree of protein binding of drugs, especially for interactions between highly bound drugs, has long been a confusing concept in pharmacokinetics. Plasma albumin concentrations range from 3.5–5 g/dl or 35 000–50 000 μg/ml, significantly greater than AAG concentrations which are 0.04–0.1 g/dl or 400–1000 μg/ml. The albumin pool is characterized by high capacity/low affinity for drugs, so drug molecules “hop on and hop off” quite readily. The AAG pool has a low capacity/high affinity for drugs. Most drugs display concentration‐independent (linear) binding to plasma proteins. Just because a drug is 99% bound to plasma proteins does not mean that 99% of the binding sites on the plasma proteins are occupied. Therapeutic drug concentrations are typically in the 1–100 μg/ml range, so the plasma protein pool capacity for binding drugs is almost never exceeded by the amount of drug in circulation. Even patients with hypoalbuminemia do not have limitations in albumin‐binding sites. The confusion in the literature comes from the misunderstanding between the free drug concentration and the free drug fraction. They are not synonymous but are frequently used interchangeably and incorrectly. The unbound drug fraction = the free drug concentration/total drug concentration. Or: In the living animal, it is the body’s clearance rate for the drug that remains constant, not the total drug concentration. When a second highly protein‐bound drug with greater affinity for a specific binding site is administered, there is potentially an extremely brief period where there is an increase in free drug concentration, but typically there is no measurable clinical effect, as the newly liberated drug molecules are either further distributed into tissues and/or the body’s clearance mechanisms kick in to eliminate them. Therefore, the unbound fraction (fu) increases, but the total drug concentration in the plasma (Ctot) decreases and the free concentration (Cfree) actually remains the same. The same thing happens when the plasma concentration of the binding protein decreases, such as with hypoalbuminemia from nephropathy. Therapeutic drug monitoring of antimicrobials in such situations finds that measured Ctot increases but is of no concern since the overall pharmacodynamic effect depends only on Cfree. In inflammation, when AAG concentrations increase, then the binding capacity of basic drugs increases. So fu decreases and Ctot increases but Cfree is still unchanged. Clinically significant drug–drug interactions from protein‐binding site displacement can only occur under very specific circumstances. The drugs must be highly protein bound (>90%), be administered intravenously, have a narrow therapeutic index and have a high hepatic extraction ratio (Rolan, 1994) (Figure 4.7). The antimicrobials used clinically in veterinary medicine do not meet these criteria, and concerns over interactions between highly protein‐bound antimicrobials (e.g., ceftiofur) and other highly protein‐bound drugs (e.g., phenylbutazone) are unwarranted. Bioavailability (F) is a measure of the systemic availability of a drug administered by a route other than intravenously (Toutain and Bousquet‐Melou, 2004a). Bioavailability is determined by comparing the area under the plasma drug concentration versus time curve (AUC) for the extravascular formulation to the AUC for the intravenous formulation. AUC is calculated by computer or by the trapezoidal method, where the entire curve is divided into trapezoids and the area of each trapezoid is calculated and summed to give the AUC (Figure 4.8). In some cases, absorption of a drug after an oral dose may be delayed due to physiological factors such as stomach emptying time and intestinal motility, causing a lag time in the graph with the entire curve shifted to the right. Figure 4.7 Algorithm for determining clinically relevant protein‐binding interactions. Source: Adapted from Rolan (1994). Figure 4.8 Plasma drug concentration profiles intravenous versus oral administration of ampicillin in a dog. Calculation of the area under the curve (AUC) for each curve is done by the trapezoidal rule. It is illustrated here for the oral curve, where the trapezoidal area between data points is calculated and summed to give the total AUC. The method of corresponding areas is used to determine F: where AUC is the total area under the plasma concentration‐time curve relating to the route of drug administration (IV versus PO, IM, or SC). The systemic availability of orally administered antimicrobial drugs is often incomplete (<100%). This may be due to poor absorption, degradation in the stomach or rumen, or presystemic metabolism (first‐pass effect). Because of species differences in digestive physiology and anatomical arrangement of the gastrointestinal tract, the bioavailability and rate of absorption of drugs administered orally differ widely between ruminant and monogastric species. Incomplete systemic availability can often be compensated for by administering a higher oral dose:
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Pharmacokinetics of Antimicrobials
Pharmacokinetic Models, Compartments, Rates, and Orders of Reactions
Zero Order
First Order
One‐compartment Open Model with IV Injection and First‐order Elimination
One‐compartment Open Model with First‐order Absorption and Elimination
Two‐compartment Open Model with IV Injection and First‐order Elimination
Multicompartment Models
Drug Distribution
Compartment
% Body Weight
Vd (l/kg)
Total body water
60–70
0.60
Intracellular water
30–40
0.30
Extracellular water
25–30
0.30
Low Vd Drugs (<0.3 l/kg)
High Vd Drugs (>1.0 l/kg)
Beta‐lactams (penicillins, cephalosporins)
Fluoroquinolones
Trimethoprim
Tetracyclines
Macrolides
Chloramphenicol
Metronidazole
Rifampin
Aminoglycosides
Medium Vd Drugs (0.3–1.0 l/kg)
Sulfonamides
Florfenicol
Lipid Solubility and Drug Ionization (pH‐Partition Hypothesis)
Acids
Penicillins
Cephalosporins
Sulfonamides
Bases
Aminoglycosides
Macrolides
Chloramphenicol/florfenicol
Trimethoprim
Amphoteric
Fluoroquinolones
Tetracyclines
Protein Binding
Bioavailability
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