Stanford University



Pharmacokinetics and Pharmacodynamics

The Anesthesiologists’ Perspective

Steven L. Shafer, M.D.

Associate Professor of Anesthesia, Stanford University

Staff Anesthesiologist, Palo Alto Veterans Administration Medical Center

Introduction:

I hated pharmacokinetics as a medical student. There were too many logarithms, exponents, and other weird stuff. Studying pharmacokinetics meant memorizing a bunch of equations. Why would anyone waste time on that? Then I became an anesthesiologist, and found I could improve the accuracy with which I can anesthetize patients if I base my drug dosing on pharmacokinetics.

You can use pharmacokinetics to reconstruct the likely plasma drug concentrations from any dose you choose to give a patient. You can use pharmacokinetics to help you decide which drug to give, and how to best dose it, to achieve any given effect. That also applies to investigators: using pharmacokinetics you can find the best way to give a new drug during a clinical trial, and thus study the drug when it is used to best advantage. This is why the FDA is now emphasizing detailed pharmacokinetic studies very early in the development of every new pharmaceutical.

You don't have to memorize a bunch of equations to understand and use pharmacokinetics. In fact, the complex equations interfere with understanding pharmacokinetics. Please don't memorize the equations that will follow. Instead, try to remember the shapes of the curves, as shown in the figures. If you find you need the equations, you can always look them up. If you like programming, or playing with spreadsheets, you can incorporate the equations into your program, and from then on treat the pharmacokinetics as a black box. This will help make pharmacokinetics fun, rather than painful and boring.

Most courses in pharmacokinetics begin with discussion of the underlying physiologic properties of the body: clearance and volumes of distribution. Because this is the standard approach, it is the one I will follow in these notes. However, starting with the physiologic basis of pharmacokinetics implies we actually learn these things from pharmacokinetic studies! Pharmacokinetic studies are almost always studies of drug concentration in the blood or plasma following known doses. As such, pharmacokinetic studies give us mathematical formulae to characterize plasma drug concentration. Using these formulae, we can predict the likely concentrations following any dose of drug. We can also use these equations to help us administer our drugs intelligently. However, physiologic conclusions, such as volume of distribution and clearance, are inferred using assumptions derived from absurdly simple models of physiology. So, view the physiologic inferences very skeptically.

I'm an anesthesiologist. I spend at least 2.5 days every week administering anesthesia in the operating room. My clinical perspective will bias my presentation in several ways:

1) Some of the drugs I will use for examples will be drugs used in the practice of anesthesia,

2) Many of the examples I use will involve intravenous administration.

3) Things happen very fast in anesthesia, and the plasma concentrations over the first 5-15 minutes are critical to the anesthetic practice. The early concentrations are predominantly a function of the distribution of drugs into peripheral tissues. For many other areas of medicine, the initial time course of the plasma concentration is almost irrelevant, and so the pharmacokinetics of distribution are not clinically important,

4) In anesthesia, we are accustomed to being highly invasive, and thus we routinely obtain high-resolution pharmacokinetic and pharmacokinetic data in clinical trials. For example, in our studies at the Palo Alto VA we generally collect arterial blood for drug assay every 30 seconds. We record measures of drug effect, such as EEG and arterial waveforms, continuously. As a result, our pharmacokinetic and pharmacodynamic data constitute nearly continuous functions of time. I think this is why my approach will focus on functions, and the shapes and relationships of functions.

The physiologic basis of pharmacokinetics and pharmacodynamics

[pic]

Figure 1: Volume of distribution represents dilution of drug into a volume

The fundamental pharmacokinetic concepts are volume and clearance. Volumes represent the apparent dilution of a drug from the concentrated form in the syringe to the far more dilute concentration floating around in the blood. It is as if the drug was pored into a bucket, as shown in figure 1. As you know, concentration is simply amount/volume. Let's say you know the amount (the dose you gave), and you measure the drug concentration in the blood. It doesn't take rocket science to rearrange the definition of concentration to solve for volume: volume = amount/concentration. From this simple relationship you can glean an important insight. Let's assume the size of the bucket is constant (a very reasonable assumption for most drugs). If you double the dose, you will double the concentration. This is the principle of “linearity.”

Clearance is the body's ability to remove drug from the blood or plasma. Clearance has units of flow: volume/time. Clearance is thus the flow of blood or plasma, expressed as volume per unit time, from which drug has been irreversibly removed, as shown in figure 2.

[pic]

Figure 2: Clearance represents the flow of blood or plasma cleared of drug

Clearance describes an intrinsic capability of the body, not an actual rate of drug removal. The rate of drug removal depends on the concentration of drug in the body. For example, if the body has a clearance of 1 liter/minute for a particular drug, the actual rate of drug removal will be 0 if no drug is present in the body, 1 mg/min if the plasma drug concentration is 1 mg/liter, 100 mg per minute if the plasma drug concentration is 100 mg/liter, etc. For drugs with linear pharmacokinetics, the clearance does not depend on the concentration of drug. The rate at which a drug is actually cleared is the product of the concentration of drug in the plasma and the clearance, as initially presented for the 1 compartmental model.

[pic]

Figure 3: The one compartment model, with a single volume and flow term.

We can combine the bucket model with the flow model, and get the classic “one compartment” pharmacokinetic model, as shown in figure 3. This model has a single volume and flow. When we get into the mathematics below, we will start with this simple model.

If we were built like buckets, we would have a single volume and a single clearance. The math would be easy, but life would be dull. For most drugs, we behave as if we were several buckets connected together by pipes, as shown in figure 4 (two compartment model) and figure 5 (three compartment model). The volume to the left in the two compartment model, and in the center of the three compartment model, is called the “central volume.” The other volumes are called “peripheral volumes,” and the sum of the all the volumes is the volume of distribution at steady state (or Vdss). The clearances leaving the central compartment for the outside is the “central” or “metabolic” clearance. The clearances between the central compartment and the peripheral compartments are called “intercompartmental clearances.”

[pic]

Figure 4: Two compartment pharmacokinetic model, with two volumes, (central and peripheral) and two clearances (central, and intercompartmental).

What do the volumes and clearances estimated by pharmacokinetic modeling mean? It is likely that the central (or “metabolic”) clearance estimated by pharmacokinetic modeling has a true physiologic basis. It is also conceivable that the volume of distribution at steady state (Vdss) has a physiological basis: the partitioning of drug into all body structures at steady state.

[pic]

Figure 5: Three compartment model, with three volumes, (central, rapidly equilibrating peripheral and slowly equilibrating peripheral) and three clearances (central, rapid and slow intercompartmental).

For three compartment models, it is tempting to speculate that the rapidly equilibrating volume (V2) corresponds to vessel rich group and the slowly equilibrating volume (V3) corresponds to the fat and vessel poor group. In fact, many authors discuss drugs in exactly this way. This may provide some insight, particularly for highly lipophilic drugs in which a large V3 may be explained by extensive distribution of the drug into fat. As we will see later on, the volumes and clearances (except central clearance and Vdss) developed in pharmacokinetic models are simply mathematical constants derived from equations that describe the plasma drug concentrations over time. The volumes and intercompartmental clearances of drugs estimated using pharmacokinetic modeling are not direct measures of anatomic structures or human physiology. This is not to imply that there is no physiological basis for the volumes and clearances that are defined in pharmacokinetic analysis. The volumes and clearances are determined by the underlying physiology, but the relationships are very complex. Below we will review the physiologic basis of the volumes and clearances, but the reader should maintain a healthy skepticism about the literal “truth” of the very simple models that will be presented.

Hepatic clearance

Many drugs are cleared by hepatic biotransformation. The extraction ratio is the ratio between the amount of drug that flows through the liver and the amount of drug extracted (i.e., cleared) by the liver. For some drugs (propofol, an intravenous anesthetic, for example), the liver removes nearly all of the drug flowing through it, resulting in an extraction ratio of 1 (i.e., 100%). For these drugs, the clearance is simply liver blood flow. Clearly, any reduction in liver blood flow will reduce clearance for drugs with high extraction ratios. Such drugs are therefore said to be “flow dependent.” Another way to think about flow dependent drugs is that the capacity of the liver to metabolize “flow dependent” drugs is way in excess of the usual flow of drug to the liver. One good aspect of flow dependent drugs is that changes in hepatic function per se will have no impact on clearance.

For many drugs, (for example, alfentanil, an intravenous opioid), the extraction ratio is considerably less than 1. For these drugs, clearance is limited by the capacity of the liver to take up and metabolize the drug. These drugs are said to be “capacity dependent.” Any change in the capacity of the liver to metabolize such drugs will affect clearance. However, changes in liver blood flow, as might be caused by the anesthetic state itself, usually have little influence on the clearance since the liver can only handle a fraction of the drug it sees anyway.

Both liver volume and liver blood flow decrease with advancing age.[i],[ii] Intrinsic hepatic metabolic capacity also decreases with age. Additionally, hepatic enzymes can be induced by other drugs or substances in the environment. Such induction can increase the clearance of capacity limited drugs (i.e., drugs with low hepatic extraction ratios). Smoking can induce liver enzymes, predominantly in young patients.46 As patients age, the liver then becomes more refractory to induction, and clearance decreases. This has been found to partly explain the reduced clearance of lorazepam in elderly individuals.[iii] Lastly, drugs themselves can alter hepatic blood flow. For example, halothane (an inhalational anesthetic) decreases liver blood flow by 60% in dogs.[iv],[v]

Renal clearance

The kidneys use two mechanisms to clear drug from the body: filtration at the glomerulus, and excretion into the tubules. Renal blood flow is inversely correlated with age,[vi] as is creatinine clearance, which can be predicted from age and weight:[vii]

Men:

[pic]

1

Women:

85% of the above.

The above equation shows that age is an independent factor in predicting creatinine clearance. Thus, elderly subjects will have decreased creatinine clearance, even in the presence of a normal serum creatinine. The decrease in renal clearance of drugs will obviously increase the concentrations and delay the offset of renally excreted drugs. From my perspective as an anesthesiologist, most of the intravenous drugs used in anesthesia practice are cleared by hepatic metabolism rather than renal excretion. The most commonly used anesthetic drug with primarily renal excretion is pancuronium, which is about 85% renally excreted.[viii] Obviously, the dosage of pancuronium must be reduced considerably in elderly patients, even in the presence of normal serum creatinine. Drugs can also affect renal blood flow. The inhalational anesthetics have been shown to decrease renal blood flow independent of their effects of cardiac output.[ix]

Distribution clearance

Distribution clearance is the transfer of drug between the blood or plasma and the peripheral tissues. It is a function of tissue blood flow, and permeability of the capillary walls to the drug. For a drug which is avidly taken up in peripheral tissues, such as propofol (a lipophilic anesthetic drug), the sum of the metabolic clearance and the distribution clearance approaches cardiac output. For drugs which are metabolized directly in the plasma, such as remifentanil (an opioid whose ester linkage is cleaved by circulating esterases), the sum of metabolic and distribution clearance can exceed cardiac output.

The tissue blood flow varies with cardiac output, which, in turn, changes with disease and in response to many drugs. Age, per se, does not reduce cardiac output in the absence of hypertension, coronary artery disease, valvular heart disease, or other cardiovascular pathology[x] although some studies have identified a small decrease associated with age.[xi] Drugs can also raise or lower cardiac output, or alter the distribution of cardiac output (i.e., redirect regional blood flow). Anesthesia decreases cardiac output, regardless of the drug administered.[xii],[xiii],[xiv],[xv],[xvi],[xvii],[xviii],[xix]

Decreases in cardiac output would be expected to decrease intercompartmental clearance. The net effect of decreased intercompartmental clearance is to increase the plasma concentrations immediately during drug administration. Following termination of drug administration, the role of decreased intercompartmental clearance is complex. However, decreased intercompartmental clearance results in a more rapid decrease in plasma concentrations following long infusions, or when a large decrease in plasma drug concentration is desired.

Central volume of distribution

[pic]

Figure 6: True and plasma concentrations following bolus injection.

The central volume of distribution is the volume that, when divided into the initial bolus dose of an intravenous drug, results in the initial concentration. As figure 6 shows, this is based on a notion that the plasma concentration instantaneously peaks, and then continuously declines.

Following an intravenous bolus of drug it is obviously false to claim that the plasma concentration has peaked at time 0. Instead, the concentration of drug in the arterial blood immediately after intravenous injection is 0. Time is required for the blood to flow from the venous to the arterial circulation. Figure 6 also shows the true time course of concentration following intravenous injection (solid line).

So, what is this initial concentration, expressed as dose / the volume of the central compartment? It is the backward extrapolation of the concentration vs time curve from its peak at about from 30 seconds to the vertical axis. It can be thought of as the initial concentration, had circulation been infinitely fast. However, a more useful way to think of dose / central volume is as the intercept for a curve that adequately describes the concentrations from roughly 30 seconds onwards, but fails miserably prior to that.

What influences the central volume? First, central volume is highly influenced by study design. A study with arterial samples (as all PK studies should ideally be performed) will have higher initial concentrations than a study with venous samples, and thus will have a smaller central volume.

[pic]

Figure 7: the influence of sample timing on estimation of central volume.

It is also influenced by the frequency of blood sampling. Figure 7 shows the backward extrapolation of the curve to the vertical axis based on samples starting at 30 seconds, and samples starting at 5 minutes. When blood sampling starts at 5 minutes, much of the initial rapid decrease in concentration is missed. Therefore, the backward extrapolation assumes a less steep slope, concludes that the initial concentration was much lower, and thus the estimate of the central volume is much larger.

Physiologically, the central volume represents the initial dilution volume into which the drug is mixed. This reflects the volume of the heart, great vessels, and the venous volume of the upper arm. It also reflects any uptake into the pulmonary parenchyma prior to the blood reaching the arterial circulation. For drugs directly metabolized in the plasma, V1 also reflects the metabolism of the drug en route from the venous cannula to the arterial sampling catheter, which appears pharmacokinetically as dilution of the drug into a larger space.

In elderly patients, the decreased total body water and redistribution of cardiac output result in a decreased central compartment volume. As this central compartment volume determines the initial plasma concentration following rapid intravenous administration, the peak concentrations in elderly individuals may be increased because of the decreased size of the central compartment, even though the steady-state distribution volume is often increased because of the increased body fat.

Peripheral volumes of distribution

The distribution volume is the volume which relates the plasma drug concentration to the total amount of drug in the body. If you could know the total amount of drug in the body, Xtotal drug, and you knew the concentration of drug in the plasma, Cplasma, then you could derive a volume term relating these:

[pic]

2

While the above equation could be calculated at any point in time following drug administration, it is most useful to think of the relationship during an infusion, after all body tissues have equilibrated with the plasma. This situation is called steady state, and at steady state Vtotal body becomes Vdss, the volume of distribution at steady state. Vdss is the algebraic sum of the peripheral volumes and the central volumes estimated by compartmental modeling.

The peripheral volumes primarily reflect the physicochemical properties of the drug that determine blood and tissue solubility. The relative solubility of a drug in blood and tissue determines how the drug partitions between the plasma and peripheral tissues. Since the peripheral volumes are determined at steady state, the flow to the tissues should not affect the size of the peripheral tissues, although tissue blood flow will certainly affect the time it takes to get to steady state.

Since these solubilities are constants, it would seem likely that volumes of distribution would change little between individuals. However, changes in body habitus and composition can occur, which will influence peripheral volumes of distribution. For example, casual observation of elderly individuals reveals several obvious physiological changes associated with age: 1) lean body mass decreases with age, 2) body fat increases with age, and 3) total body water decreases with age.[xx] These changes in body habitus and muscle and fat distribution might be expected to produce changes in the volumes into which drugs distribute. For example, in a study of several benzodiazepines, lipid solubility was found to predict the volume of distribution (Vd).[xxi] Since the lipid content of elderly patients is higher than that of young patients, one might reasonably expect that elderly patients would have larger volumes of distribution. Increased volume of distribution, with increased duration of drug effect, has been documented in elderly individuals for trazodone[xxii] and nitrazepam.[xxiii] However, as mentioned before, the partitioning of drugs into body tissues is dictated by physicochemical properties of the drug itself, and thus the partitioning, per unit of tissue, is not influenced by age.[xxiv]

Protein binding

Protein binding affects both volumes and clearances. Protein binding can have an important influence on pharmacokinetics, because protein binding is affected by many diseases and also changes with age.

Albumin and alpha 1-acid glycoprotein are the primary sites of protein binding. Albumin concentration decreases with advancing age, hepatic disease, and malnutrition. In contrast, alpha 1-acid glycoprotein concentration increases with advancing age, and also with acute disease. Thus, the effects of age and disease on protein binding depend on which protein binds the drug.

Let us consider the in vitro influence of changes in plasma proteins. For drugs which are highly protein bound, a change in protein concentration results in a nearly inversely proportional change in the concentration of free drug in the plasma. For drugs with minimal protein binding, a change in protein concentration produces a minimal change in the free concentration. Most of the intravenous drugs used in anesthetic practice are highly protein bound.

The in vitro observation that changing protein concentration results in changing free drug concentration doesn't necessarily apply to the in vivo situation. It is the free (e.g., unbound) drug that equilibrates between the plasma and the tissues. If protein binding is decreased, then the driving concentration gradient increases between the plasma and the peripheral tissues. As a result, when protein binding decreases, equilibrium is achieved between the plasma and the tissue free drug concentrations at a lower total plasma drug concentration. This lower concentration gives the appearance that the drug has distributed into a larger total space. Thus, decreased protein binding causes an increase in the apparent volume of distribution.

However, for lipophilic drugs the free drug concentration in the plasma is mostly determined by equilibration with other tissues, not by equilibration with plasma proteins. Thus, the free drug concentration in the plasma is not highly affected by changes in protein binding for lipophilic drugs in vivo following equilibration with peripheral tissues.

The increase in Vdss seen with decreased plasma protein binding is mostly an illusion, caused by referencing the drug to the total, rather than the unbound, plasma drug concentration. Were only the unbound drug concentration measured, then for lipophilic drugs (such as most of those in anesthesia practice), there would be almost no change in the apparent volume of distribution, since the concentration of free drug in the plasma after equilibration with peripheral tissues is only trivially affected by bound drug in the plasma.

Changes in protein binding may also affect the clearance of drugs. If a drug has a high extraction ratio. then the liver is going to remove nearly all of the drug flowing to it, regardless of the extent of protein binding. However, if the drug has a low hepatic extraction ratio, then an increase in the free fraction of drug will result in an increase in the driving gradient, with an associated increase in clearance.

Lastly, protein binding also affects the apparent potency of a drug, when referenced to the total plasma drug concentration. An increase in free fraction increases the driving pressure to the site of drug effect (discussed several pages hence), and thus increases the concentration in the effect site. Thus, decreased protein binding may decrease the dose required to produce a given drug effect even in the absence of pharmacokinetic changes.

As mentioned above, albumin concentrations decrease with age, while alpha 1-acid glycoprotein increases with age Since diazepam primarily binds to albumin, the free fraction increases in elderly patients, and this has been shown to correlate with reduced dose requirements,[xxv],[xxvi] probably from an apparent increase in steady-state potency from the increased free fraction. In contrast, lidocaine binds primarily to alpha 1-acid glycoprotein, and in elderly patients increased alpha 1-acid glycoprotein reduces the free fraction, which may contribute to the reduced clearance.66 Phenytoin binds to albumin, and hence phenytoin clearance is inversely correlated with plasma albumin. Thus, phenytoin clearance actually increases with age.[xxvii]

Stereochemistry

The last PK/PD concept to introduce is that most of the analysis presented describe fictitious drugs: thiopental, fentanyl, midazolam, etc. Most drugs are chiral, and are supplied as racemic mixtures. There is no reason to believe that the pharmacokinetics and pharmacodynamics of the enantiomers are identical. The body is a chiral environment, and thus how drugs interact with receptors, enzymes, proteins, is stereospecific. When a racemic mixture is given, it is as if two different drugs have been infused. However, drug assays are usually insensitive chirality. Thus, the concentration measured is actually a mixture of two separate drugs, each of which may have unique PK and PD characteristics.

Mather and colleagues have studied the PK and PD of the enantiomers of bupivacaine,[xxviii] mepivacaine,[xxix] and prilocaine.[xxx] The enantiomers of ketamine also have been extensively studied,[xxxi],[xxxii] in the hope that the S+ isomer will provide the hypnosis and analgesia associated with ketamine without the undesirable psychotomimetic side effects. Although importance of studying the individual PK and PD of stereoisomers is appreciated,[xxxiii] the difficulty of doing such studies has precluded more widespread analysis.

Mathematical principles

Many processes in life happen at a constant rate, like the power consumption of a clock, or the rate at which we age. These processes are called zero-order processes. The mathematics of the rate of change (dx/dt) are simple for zero-order processes:

rate (dx/dt) = k (a constant)

The units of k are amount/time. If we want to know the value of x at time t, x(t), we can solve it as the integral of the above equation from time 0 to time t:

x(t) = x0 + kt

where x0 is the value of x at time 0. This is, of course, the equation of a straight line with a slope of k and an intercept of x0.

Other processes occur at a rate proportional to the amount. For example, the rate at which we pay interest on a loan is proportional to the outstanding balance. The banker doesn't say: “You'll pay $25 in interest every month, no matter how much you borrow.” (My response would be: “fine, I'll borrow a million dollars.”) Instead, he says: “You'll pay 1% of the outstanding principle every month.” This is an example of a first order process. Compared to a zero-order process, the mathematics of a first-order process are modestly more complex. The rate of change for a first-order process is:

rate (dx/dt) = kx.

Here, the units of k are simply 1/time, since the x to the right of the “=” already brings in the units for the amount. If we want to know the value of x at time t, x(t), we can solve it as the integral of the above equation from time 0 to time t:

x(t) = x0 ekt

where x0 is the value of x at time 0. If kt > 0, x(t) increases exponentially. If kt > 0, x(t) decreases exponentially. In pharmacokinetics, the exponent is negative, i.e. concentrations decrease over time. However, to simplify the upcoming calculations, we will express the relationship between x and t by removing the minus sign from k, and expressing it explicitly in the equation. Thus, the equation we will explore will be:

x(t) = x0 e-kt

[pic]

Figure 8: x = x0 e-kt

Figure 8 is a graph showing the relationship between x and time, as described by the above equation (where k is positive, so -kt is < 0). Such a graph might describe the plasma drug concentrations after rapid intravenous injection (called a “bolus”). Note how the concentrations continuously decrease, and the slope continuously increases, as the levels fall from x0 to 0, which is approached as t → ∞.

If we take the natural logarithm* of both sides of the above equation, we get:

ln(x(t)) = ln(x0 e-kt)

= ln(x0) + ln(e-kt)

= ln(x0) - kt

[pic]

Figure 9: ln(x) = ln(x0) - k t

This is the equation of a straight line, where the vertical axis is ln(x(t)), the horizontal axis is t, the intercept is ln(x0) and the slope of the line is -k. This is why people like to graph first order processes using the logarithm of the value vs time: the graphs become straight lines, as shown in figure 9.

How long will it take for x to go from x0 to x0/2, i.e., for the x to fall by 50%? We can relate the slope of the line (-k) to the change in x and t as follows:

[pic]

3

where t ½ is the time required for a 50% decrease in x.

We can simplify the numerator to:

[pic]

4

This neatly relates the slope, k, to the time required for a 50% change, t ½:

[pic]

5

so if we measure t ½, the time it takes for x to fall by 50%, we then know the exponent, k, as calculated above. If we know k, the exponent, then the time it will take for x to fall by 50% is simply:

[pic]

6

Thus, exponential functions are intrinsic to solving for the amount, x, at time t, when dealing with first order processes, and logarithms are useful to transform the exponential curve into a straight line, which can then be more easily manipulated.

[pic]

Figure 10: The one-compartment model.

Let's now return to the simplest pharmacokinetic model, the tried and untrue “one compartment model,” as shown in figure 10. However, this time we will combine the model with some of the above mathematics. Let's assume that we are built like a fluid filled bag, into which an amount of drug is injected. The concentration of drug in the bag, C, is simply the amount of drug present, x, divided by the volume of the bag, V.

Let's say that fluid flows out of the plastic bag at a rate, Q. This is the same as the metabolic clearance previously described, but I'll call it Q here to emphasize that we are dealing with flow. The rate at which drug (x) flows out is thus the rate of fluid flow, Q, times the concentration of drug in the fluid, C. Thus, the rate at which drug flows out of the bag is:

[pic]

7

This is a first-order process. We can find k, the rate constant, by substituting x/V for C in the above equation:

[pic]

If Q/V x = k x, k must equal Q/V. Rearranging this yields a fundamental pharmacokinetic statement:

Q (clearance) = k (rate constant) × V (volume of distribution)

This leads to two useful insights: If the volume remains constant, then as Q (clearance) increases, k increases, and the half-life decreases. If the clearance remains constant, then as V (volume) increases, k decreases, and the half-life increases. If we know the flow out of the compartment (clearance), and we know the volume of the compartment, we can calculate k as Q/V. We can then calculate the half-life of drug in the bag as 0.693/k.

Let A = x0/V, where A is the concentration at time 0, x0 is the initial dose of drug and V is the volume of the bag. The plasma concentrations over time following an intravenous bolus of drug are then described by an equation of the form:

C(t) = Ae -kt.

This is the commonly used expression relating concentration to time and initial plasma concentration, and the rate constant. It defines the “concentration over time” curve for a 1 compartment model, and has the log linear shape seen in figure 9.

We can calculate the flow, Q, in one of two ways. First, as noted above, if we know V and k, then Q = k V. However, a more general solution is to consider the integral of the concentration over time curve, known in pharmacokinetics as the “area under the curve” or “AUC” for short. This integral can be solved as:

[pic]

8

Thus, Q = x0 /AUC, showing that the clearance equals the dose divided by the area under the curve. This is a fundamental property of linear pharmacokinetic models. It directly follows that AUC is proportional to dose for linear models (i.e., models where Q is constant).

Let's say that you start giving an infusion at a rate of I (for Input) to a person who has no drug in his or her body. It's pretty obvious that the plasma concentration will continue to rise as long as the rate of drug going in the body, I, exceeds the rate at which drug leaves the body, C × Q. Once, I = C × Q, drug is going in and coming out at the same rate, and the body is at steady state. This raises two questions: 1) what is the eventual concentration? and, 2) how long will it take until I = C × Q?

To answer the first question, just consider that when the body is in equilibrium, the rate of drug going in must equal the rate of drug coming out, and thus I (the rate of drug going in)= C (the steady state concentration) × Q (clearance). Without rocket science we can rearrange that equation to tell us that C = I/Q, the ratio of the infusion rate and the clearance. Thus, the eventual concentration is the rate of drug input divided by the clearance.

C = I/Q is satisfyingly similar to the equation describing the concentration following a bolus injection: C = x0 / V. This suggests another way to think about volume and clearance: volume relates initial concentration to the size of the initial bolus, and clearance relates steady-state concentration to the infusion rate. One consequence is that the initial concentration following a bolus is independent of the clearance, and the steady state concentration during a continuous infusion is independent of the volume.

As far as how long it will take to reach steady state, the answer is simple: ∞. The reason is that the steady state concentration is asymptotically approached, but never reached. However, we can determine how long it will take to reach any given fraction of the steady state concentration. The rate of change in x, the amount of drug in the compartment, is:

[pic]

9

where I is the rate of drug going in, x(t) is the amount of drug present at time t, and k x(t) is the rate of drug coming out (this is a first order process, right?). To find x(t), we need to integrate this from time 0 to time t, knowing that x(0)=0 (i.e., we are starting with nothing in the body). If we integrate this over time to find x at any time t, we get:

[pic] Equation 1

As t →∞, , e-kt → 0, and the above equation reduces to:

[pic] Equation 2

Let's say that we want to get to 50% of that amount, i.e., x(∞)/2. From the above equation, we know that x(∞)/2 = I/(2k). Substituting I/(2k) for x(t) into equation 1, we get:

[pic]

10

Solving this for t, we get: ln(2)/k. As you may recall, we previously showed that the half-life, t ½, following a bolus injection was ln(2)/k. Here we again have a satisfying parallel between boluses and infusions: with an infusion, the time to get to 50% of the steady state concentration is 1 half-life. We can similarly show we will get to 75% of the steady state concentration following 2 half-lives, 88% following 3 half-lives, 94% following 4 half-lives, and 97% following 5 half-lives. Usually, by 4-5 half-lives, we consider the patient to be at steady state, although they are a few percent (but an infinite time!) away from truly being at steady state.

One peripheral observation. Let's again return to equation 2 above. Remembering that k = Q/V, we get x(∞) = I V / Q. By definition, C(∞) = x(∞)/V. If we solve equation 2 for C(∞), we get:

[pic]

Thus, we have again shown that the steady state concentration is I/Q. However, I prefer the more intuitive approach at the top of page 15 to demonstrate the relationship between concentration at steady state, infusion rate, and clearance.

Let's summarize the essential concepts developed so far:

1. The rate of change (decrease) when drug is injected into a 1 compartment model is:

dx/dt = -kx (first order process).

2. The concentration following that injection is:

x(t) = x0 e-kt

where x0 is the initial dose.

3. The half-life, t ½, (time required for a 50% decrease) is:

t ½ = 0.693/k.

4. If you know the time required for a 50% decrease, the rate constant, k is:

k = 0.693/(t ½)

5. The definition of concentration is:

C = x (amount) / V (volume)

6. The concentration following the bolus will be:

C(t) = x0/V e-kt

where x0/V = initial concentration following bolus.

7. If Q is the flow (clearance) from a 1 compartment model, the rate at which drug leaves can be calculated:

dx/dt = - C × Q

(as in equation 1, we add the minus so we can express clearance as a positive number.)

8. Since item 1 above, and item 7 above, are the same rate, it follows (after substituting x/V for C) that:

k = Q/V

9. As clearance, Q, increases, k increases, and the half-life decreases. As volume increases, k decreases, and half-life increases.

10. During an infusion at rate I, the concentrations are described by the equation:

x(t) = I/k (1 - e-kt)

11. The steady-state concentration will be:

C(∞) = I/Q

12. Half-lives describe the time for a 50% decrease in concentration following a bolus, and the time required to reach 50% of the steady state concentration during an infusion. Following a bolus, the concentrations will be at 25%, 13%, 6%, and 3% of the initial concentration following 2, 3, 4, and 5 half-lives, respectively. During an infusion, the concentration will reach 75%, 88%, 94%, and 97% of the steady state concentration in 2, 3, 4, and 5, half-lives, respectively.

What do you do with this? Well:

1) If you know the amount of drug injected, x0, and the concentration at time 0, A, you can calculate the volume: V = x0 / A.

2) If you know the amount of drug injected, x0, the volume, V, and k, then you can calculate the concentration at any given time, t, as

[pic]

11

3) If you know two separate concentrations, C(t1) and C(t2), you can calculate k as:

[pic]

12

4) If you want to know the flow out of the compartment, you can calculate it as kV, if k and V are known. If k and V are not known, or, as shown below, there are several values of k, you can still calculate clearance as dose / AUC.

5) If you know the initial concentration you want to achieve, T (target), then you can calculate the intravenous dose required to produce that concentration, x0, as T × V.

6) If you want to maintain concentration T, then you must continuously infuse T at the same rate it is leaving. Assuming that you first gave a bolus of T × V, the rate at which drug will leave will be T × Q. Therefore, your maintenance infusion rate must also be T × Q.

Let's take a hypothetical example of a new drug, cephprololopam, (an antibiotic that has beta blocking and anxiolytic properties):

1) The clearance of cephprololopam is 0.2 liters/minute.

2) The volume of distribution of cephprololopam is 20 liters.

3) The therapeutic level is 2 μg/ml.

1) What is the half-life of cephprololopam?

Answer: k = Q/V = 0.2 l/min / 20 l = 0.01 min-1.

t ½ = 0.693/k = 69 minutes.

2) What is the initial dose of cephprololopam?

Answer: x0 = Ctarget × V = 2 μg/ml × 20 liters = 40 mg.

3) What cephprololopam infusion is needed to maintain a cephprololopam concentration of 2 μg/ml?

Answer: I = Ctarget × Q (clearance) = 2 μg/ml × 0.2 l/min = 0.4 mg/min.

4) What is the oral dose of cephprololopam, taken every 24 hours, to maintain a target of 2 μg/ml, assuming complete absorption?

Answer: 0.4 mg/min × 1440 minutes = 576 mg/day.

5) How long will it take to reach steady state dosing with these repeated oral doses (ignoring the time course of absorption)?

Answer: 4 to 5 half-lives = 276 to 345 minutes, (the patient will be at steady state dosing within the time course of the first dose.)

We have focused so far on intravenous drug delivery, which reflects my own therapeutic orientation as an anesthesiologist. However, question 5 was designed to show that the same concepts apply to oral dosing, with some minor modifications. With intravenous dosing, all of the drug reaches the systemic circulation. For oral dosing, before the drug reaches the systemic circulation it must pass through the liver, which will metabolize some of the ingested drug. Thus, the total drug reaching the circulation is not the administered dose, but the administered dose times F, the fraction “bioavailable.” Question 5 tacitly assumed that cephprololopam was 100% bioavailable. If the bioavailability was less than 100%, then the administered dose would have been:

[pic]

where F is the fraction bioavailable.

Also, with oral delivery, the drug is absorbed from the intestinal track. The input to the systemic circulation is initially high, and declines over time as the drug gets absorbed through the gut and, possibly, degraded in the intestinal tract. This is often modeled as a monoexponential decrease in delivered drug, so that the rate of input is not a constant infusion, I, but rather an infusion that changes with time:

[pic]

where F is the fraction bioavailable, Doral is the dose given oral, and ka is the absorption rate constant. Note that the integral of [pic] is 1, so that the total input is F × Doral. As before, the trick to finding the concentration over time is to first express the differential equations, and then integrate. The differential equation for the amount, x, with oral absorption into a 1 compartment disposition model is:

[pic]

This is simply the rate of input at time t, I(t), minus the rate of exit, k × x. To solve for the amount of drug, x, in the compartment at time t, we integrate this from 0, to time t, knowing that x(0) = 0:

[pic]

This takes care of most of the standard PK equations for one compartment models, (nobody will ever ask you for the oral dosing equation just shown). Unfortunately for the makers of cephprololopam there are almost no drugs described by one compartmental models. As you will see, focusing on 1 compartment models, where the math is tractable, can be misleading when we consider the majority of drugs (at least in anesthesia) described by multicompartment models. We now consider the models that describe the drugs we use in anesthesia.

[pic]

Figure 11: Shape of curve following intravenous bolus injection.

The plasma concentrations over time following a bolus of an intravenous drug actually resemble the curve in figure 11. This curve has the characteristics common to most drugs when given by intravenous bolus. First, the concentrations decline over time. Second, the rate of decline is initially steep, but continuously becomes less steep (i.e., the slope continuously increases), until we get to a portion which is “log-linear.” During this log-linear phase, the slope continues to decrease over time. However, if we plot the log of concentration against time, the curve will be linear.

For many drugs, three distinct phases can be distinguished, as suggested by Figure 11. There is a rapid “distribution” phase (solid line in figure 11) that begins immediately after the bolus injection. This phase is characterized by very rapid movement of the drug from the plasma to the rapidly equilibrating tissues. Often there is a slower second distribution phase (dashed line in figure 11) that is characterized by movement of drug into more slowly equilibrating tissues, and return of drug to the plasma from the most rapidly equilibrating tissues (i.e, those that reached equilibrium with the plasma during phase 1). The terminal phase (dotted line in figure 11) is a straight line when plotted on a semilogarithmic graph. The terminal phase is often called the “elimination phase” because the primary mechanism for decreasing drug concentration during the terminal phase is drug elimination from the body. The distinguishing characteristic of the terminal elimination phase is that the relative proportion of drug in the plasma and peripheral volumes of distribution remains constant. During this “terminal phase” drug returns from the rapid and slow distribution volumes to the plasma, and is permanently removed from the plasma by metabolism or renal excretion.

Curves which continuously decrease over time, with a continuously increasing slope (i.e., curves that look like figure 11), can be described by a sum of exponentials. In pharmacokinetics, one way of notating this sum of exponentials is to say that the plasma concentration over time is:

[pic] Equation 3

where t is the time since the bolus, C(t) is the drug concentration following a bolus dose, and A, α, B, β, C, and γ are parameters of a pharmacokinetic model. A, B, and C are called coefficients, while α, β, and γ are called exponents or, occasionally, hybrid rate constants. I prefer the term “exponents,” because that's exactly what they are. Following a bolus injection all 6 of the parameters in equation 3 will be greater than 0.

So, why use equation 3 and deal with exponents? The first, and crucial, reason to use polyexponentials is that equation 3 describes the data. Why does it describe the data? Because the data resemble figure 11, in that the concentration following a bolus is always decreasing, with a slope that is increasing. Such a curve can always be described as a sum of negative exponentials. Thus, pharmacokinetics is an empirical science: the models describe the data, not the processes by which the observations came to be. The second reason to use polyexponential functions to describe the concentrations over time is that polyexponential models permit us to use many of the 1 compartment ideas just developed, with some generalization of the concepts. The third reason is that equation 3 can be mathematically transformed into a model of volumes and clearances that has a nifty, if not necessarily accurate, physiologic flavor. Lastly, equation 3 has some nice mathematical properties (for example, the integral of equation 3 is A/α + B/β + C/γ).

[pic]

Figure 12: Figure 11 as sums of exponentials

Equation 3 really says that the concentrations over time are the algebraic sum of three separate functions, Ae-αt, Be-βt, and Ce-γt. We can graph each of these functions out separately, as well as their superposition (i.e. algebraic sum at each point in time) as shown in figure 12. At time 0 (t = 0), equation 3 reduces to:

[pic]

13

The sum of the coefficients A, B, and C, equals the concentration immediately following a bolus. Usually, but not always, A > B > C. Thus, the initial contribution to the decrease in concentration, as shown in figure 12, is primarily from the α component, because Ae-αt >> Be-βt >> Ce-γt.

The exponents usually differ in size by about an order of magnitude. There are several conventions to the exponential terms. I prefer to order the exponentials as α > β > γ. However, for historical reasons some individuals always call the smallest exponent β. It is usually clear from the context which exponent is the smallest. Instead of calling them α, β, and γ, some individuals refer to them as λ1, λ2, and λ3.

There is a special significance to the smallest exponent. Let λ represent the exponents, α, β, and γ. As t → ∞, λt → ∞, and e-λt → 0. Now, as t → ∞, λt will go to ∞ the most slowly for the smallest value of λ, and hence e-λt will go to 0 the most slowly for the smallest value of λ (i.e., γ as notated in equation 3). After enough time has passed (approximately t > (log(C) - log(B))/(β- γ), if you're curious), the values of Ae-αt and Be-βt are so close to 0, relative to the value of Ce-γt that the drug concentrations over time are pretty much following the concentrations predicted just by Ce-γt, which will appear to be a straight line, with a slope of γ, when plotted as log concentration vs time.

When the pharmacokinetics have multiple exponents, each exponent is associated with a half-life. Thus, a drug described by three exponents has three half-lives, two rapid half lives, calculated as 0.693/α and 0.693/β, and a terminal half-life (sometimes called the “elimination half-life”), calculated as 0.693/γ. In the literature you will often read about the half-life of a drug. Unless it is stated otherwise, the half-life will be the terminal half-life, i.e., 0.693/smallest exponent. With all of these half-lives, you might think that it would be hard to intuit what happens when you stop giving a drug. That is absolutely correct. As pointed out by Shafer and Varvel[?] and Hughes et al,[?] the terminal half-life for drugs with more than 1 exponential term is nearly uninterpretable. The terminal half-life may nearly describe, or tremendously overpredict, the time it will take for drug concentrations to decrease by 50% after drug administration. The terminal half-life places an upper limit on the time required for the concentrations to decrease by 50%. Usually, the time for a 50% decrease will be much faster than that upper limit.

Constructing these pharmacokinetic models represents a trade-off between accurately describing the data, having confidence in the results, and mathematical tractability. Adding exponentials to the model usually provides a better description of the observed concentrations. However, adding more exponential terms usually decreases our confidence in how well we know each coefficient and exponential, and greatly increases the mathematical burden of the models. This is why most models are limited to two or three exponential terms.

As mentioned, part of the continuing popularity of polyexponential models of pharmacokinetics is that they can be mathematically transformed from the admittedly unintuitive exponential form shown above to a more easily intuited compartmental form, as shown in figures 4 and 5. The fundamental parameters of the compartment model are the volumes of distribution (central, rapidly and slowly equilibrating peripheral volumes) and clearances (systemic, rapid and slow intercompartmental). The central compartment (compartment 1) represents a distribution volume and includes the rapidly mixing portion of the blood and the first-pass pulmonary uptake. The peripheral compartments are composed of those tissues and organs showing a time course and extent of drug accumulation different from that of the central compartment. In the 3-compartment model, the two peripheral compartments may very roughly correspond to splanchnic and muscle tissues (rapidly equilibrating) and to fat stores (slowly equilibrating). The sum of the compartment volumes is the apparent volume of distribution at steady state (Vdss) and is the proportionality constant relating the plasma drug concentration at steady state to the total amount of drug in the body. If drugs are highly soluble in body tissues, (e.g., lipophilic drugs), then there will be a large amount of drug in the body, relative to the plasma drug concentration. In this situation, the Vdss will be much larger than the patient. It is not uncommon for anesthetic drugs to have Vdss of 400-500 liters, and the Vdss of propofol is in the area of 2000-4000 liters! This just means that the drug would need to be diluted by 4000 liters of plasma to get the same plasma concentration as that observed when propofol is diluted into 1 human.

“Micro rate constants,” expressed as kij, define the rate of drug transfer from compartment i to compartment j, just as k did for the 1 compartment model. Compartment 0 is a compartment outside the model, so k10 is the micro rate constant for those processes acting through biotransformation or elimination that irreversibly remove drug from the central compartment. The intercompartmental micro rate constants (k12, k21, etc.) describe the exchange of drug between the central and peripheral compartments. Each compartment has at least two micro-rate constants, one for drug entry and one for drug exit. The micro-rate constants for the two and three compartment models can be seen in figures 4 and 5. The differential equations describing the rate of change for the amount of drugs in compartments 1, 2, and 3, follow directly from the micro-rate constants (note the similarity to the 1 compartment model).

[pic]

14

where I is the rate of drug input. An easy way to model pharmacokinetics is to convert the above differential equations to difference equations, so that dx becomes Δx, and dt becomes Δt. With a Δt of 1 second, the error from linearizing the differential equations is less than 1%. Current PC's and Macs can simulate hours worth of pharmacokinetics in a matter of seconds, and you can set this up on a spreadsheet. This absurdly simple method of pharmacokinetic simulation has a name: Euler's numeric approximation. In anesthesia it has become very popular among clinicians to set up the above differential equations in spreadsheets and solve for concentrations of anesthetics drugs in the different compartments over time, but, I'm digressing...

For the one compartment model, k was both the rate constant and the exponent. For multicompartment models, the relationships are far more complex. The interconversion between the micro-rate constants and the exponents becomes exceedingly complex as more exponents are added, because every exponent is a function of every micro-rate constant and vice versa.

The offset of drug affect

In anesthesia, the offset of drug effect governs awakening from the anesthetic state. Thus, anesthesiologists are particularly concerned with the rate of decrease in plasma concentration following drug administration. Earlier, I pointed out that the terminal half-life sets an upper limit on how long it will take the plasma concentrations to fall by 50%. For drugs described by multicompartmental pharmacokinetics, the actual time for the plasma concentrations to fall by 50% is always faster than that, and often much faster.

The rate at which drug decreases is dependent both on elimination and distribution of the drug from the central compartment. Drug that has distributed into peripheral tissues is partly sequestered from the plasma, in that a gradient must be established between the concentration in the central compartment and in the peripheral tissues plasma before a net flow will be established between the peripheral tissues and the plasma. The contribution of redistribution, elimination, and sequestration towards the rate of decrease of drug concentration varies according to the duration of drug delivery. As a result the time for the drug concentration to decrease a set percentage varies according to the duration of drug administration.

[pic]

Figure 13: Context sensitive half-times (vertical axis) for sufentanil and alfentanil, as a function of infusion duration (horizontal axis).

Because half-lives tell us almost nothing about the time required for the concentrations to fall by 50%, Hughes et al introduced the term “context-sensitive half-time” to describe the time required for a 50% decrease in plasma concentration following infusions of varying duration.35 The context is the duration of an infusion that maintains a steady drug concentration. Figure 13 shows the context sensitive half-times for two opioids popular in anesthesia practice: alfentanil, and sufentanil. The terminal half-lives for these drugs are 2 hours and 9 hours, respectively. Even though sufentanil has terminal half-life that is nearly 5 times longer than that of alfentanil, the sufentanil concentrations will fall much faster than the alfentanil concentrations for infusions of less than 8 hours duration. This illustrates that terminal half-lives are misleading: the 9 hour terminal half-life of sufentanil provides virtually no insight (actually, it is misleading) into how long it will take the plasma concentrations to fall by 50% following drug administration.

[pic]

Figure 14: The times for a 20%, 50%, and 80% decrease in sufentanil concentration, as a function of infusion duration.

The clinical setting determines the percent decrease necessary to produce a given change in drug effect. Again, to use an example from anesthesia, let's say that we are running a very light anesthetic. Let us postulate that the anesthetic is so light that just a a 20% decrease in opioid concentration would produce emergence from anesthesia. Conversely, let's say that we are running a very deep anesthetic (e.g., an opioid anesthetic for cardiac surgery). In this case, we may need an 80% decrease in concentration for the patient to awaken. Figure 14 shows the times required for the sufentanil concentration to fall by 20, 50, and 80% as a function of the infusion duration. We can infusion sufentanil for 10 hours, and still et a 20% decrease within a few minutes of turning off the infusion. However, a 50% decrease will take an hour after a 10 hour infusion. If we need an 80% decrease, then after just 3 hours of drug administration we will need another 3 hours for the patient to awaken!

One cannot predict the shapes of these curves a priori. Only with computer simulations can we predict the time course of recovery following administration of drugs described by multicompartmental pharmacokinetics, which includes almost all anesthetic drugs.

Plasma-Effect Site Equilibration

[pic]

Figure 15: The plasma (solid) and biophase concentrations (dashed lines) following a bolus of 3 common opioids.

Although the plasma concentration following an intravenous bolus peaks nearly instantaneously, no anesthesiologist would induce a patient with an intravenous bolus of a hypnotic and immediately intubate the patient. The reason, of course, is that although the plasma concentration peaks almost instantly, additional time is required for the drug concentration in the brain to rise and induce unconsciousness, as shown in figure 15. This delay between peak plasma concentration and peak concentration in the brain is called hysteresis. Hysteresis is the clinical manifestation of the fact that the plasma is usually not the site of drug action, only the mechanism of transport. Drugs exert their biological effect at the “biophase,” also called the “effect site,” which is the immediate milieu where the drug acts upon the body, including membranes, receptors, and enzymes.

The concentration of drug in biophase cannot be measured. First, it is usually inaccessible, at least in human subjects. Second, even if we could take tissue samples, the drug concentration in the microscopic environment of the receptive molecules will not be the same as the concentration grossly measured in, say, ground brain or CSF. Although it is not possible to measure drug concentration in the biophase, using rapid measures of drug effect we can characterize the time course of drug effect. Knowing the time course of drug effect, we can characterize the rate of drug flow into and from the biophase. Knowing these rates, we can characterize the drug concentration in the biophase in terms of the steady state plasma concentration that would produce the same effect. This requires us to add to our model an effect compartment, as shown in figure 16.

[pic]

Figure 16: The compartmental model, now with an added effect site. ke0 is often directed outside, as though drug were eliminated from the effect site.

The effect site is the hypothetical compartment that relates the time course of plasma drug concentration to the time course of drug effect, and ke0 is the rate constant of drug elimination from the effect site. By definition the effect compartment receives such small amounts of drug from the central compartment that it has no influence on the plasma pharmacokinetics.

If a constant plasma concentration is maintained then the time required for the biophase concentration to reach 50% of the plasma concentration (t ½ ke0) can be calculated as 0.693 / ke0. Following a bolus dose, the time to peak effect site concentration is a function of both the plasma pharmacokinetics and ke0. For drugs with a very rapid decline in plasma concentration following a bolus (e.g., adenosine, with a half-life of several seconds), the effect site concentration will peak within several seconds of the bolus, regardless of the ke0. For drugs with a rapid ke0 and a slow decrease in concentration following bolus injection (e.g., pancuronium), the time to peak effect site concentration will be determined more by the ke0 than by the plasma pharmacokinetics. ke0 has been characterized for many drugs used in anesthesia.[?],[?],[?],[?],[?],[?],[?] Equilibration between the plasma and the effect site is rapid for the thiopental,9 propofol,13 and alfentanil,11 intermediate for fentanyl11 and sufentanil12 and the nondepolarizing muscle relaxants,[?] and slow for morphine and ketorolac.

[pic]

Figure 17: The plasma and effect site concentrations for propofol, assuming a t ½ ke0 of 1, 2.8 (the real value) and 5 minutes.

Using the intravenous hypnotic propofol, we can consider the influence of ke0 on the onset of drug effect. Figure 17 shows the plasma concentrations and apparent biophase concentrations after an IV bolus of propofol for three values for t ½ ke0: 1 min, 2.8 min (the actual value for propofol),[?] and 5 min. Regardless of the value of ke0, the pattern remains the same. The plasma concentration peaks (nearly) instantly and then steadily declines. The effect site concentration starts at 0 and increases over time until it equals the (descending) plasma concentration. The plasma concentration continues to fall, and after that moment of identical concentrations, the gradient between the plasma and the effect site favors drug removal from the effect site and the effect site concentrations decrease.

Examining the different values of t ½ ke0 in figure 17 shows that as t ½ ke0 increases, the time to reach the peak apparent biophase concentration also increases. Concurrently, the magnitude of the peak effect site concentration relative to the initial plasma concentration decreases because slower equilibration between the plasma and biophase allows more drug to be distributed to other peripheral tissues.

Let us again turn to two opioids used in the practice of anesthesia: alfentanil, and sufentanil. As shown in figure 15, the rapid plasma-effect site equilibration (large ke0) of alfentanil causes the effect site concentration to peak about 90 seconds after bolus injection. At the time of the peak about 60% of the alfentanil will have distributed into peripheral tissues or been eliminated from the body. For sufentanil, the effect site peaks 5-6 minutes after the bolus. At the time of the peak, over 80% of the initial bolus of sufentanil will have been distributed into the tissues or eliminated. As a result of the slower equilibration with the effect site, relatively more sufentanil than alfentanil must be injected into the plasma, which slows the rate of drug offset from a sufentanil bolus compared to an alfentanil bolus.

[pic]

Figure 18: The plasma and effect site concentrations following a bolus or infusion of propofol.

Figure 18 shows the plasma concentrations and the apparent biophase concentrations after a bolus and 10 min infusion of propofol. The degree of disequilibrium is less after an infusion than after a bolus. Thus, during an infusion the observed drug effect parallels the plasma drug concentration to a greater extent than after a bolus.

[pic]

Figure 19: The effect site concentrations over time for doses of fentanyl that target the high and low edges of the therapeutic window.

Let us now integrate our the sigmoidal relationship between concentration and effect, the concept of the therapeutic window, and the equilibration delay between the plasma and the site of drug effect. One must first identify, preferably with full concentration-response relationships, the edges of the therapeutic window. One can then produce dosage regimens that produce concentrations within the therapeutic window at the site of drug effect. The technique will be discussed below for intravenous drugs. One need also consider the rate of onset, as a small dose will result in a slower onset to a given effect than a larger dose. However, a larger dose will be more likely to exceed the therapeutic window, and possible cause toxic effects. For example, figure 19 shows the biophase concentrations after two different bolus doses of fentanyl designed to achieve the high and low edges of the therapeutic window for supplementing an induction with thiopental. The larger dose of fentanyl produces a rapid onset of drug effect and adequate biophase concentrations for a several minutes. The biophase concentration from the lower dose momentarily brushes against the lower edge of the therapeutic window.

Designing dosing regimens

Now that we have reviewed the basics of pharmacokinetics and the mathematical models, it is time to ask: how do we actually calculate drug dosages? Again, I use anesthetic drugs as examples, because these are the drugs with which I am the most familiar.

Initial bolus dose

Let's start by computing how to give the first dose of intravenous drug (although the same concepts apply to giving the first dose of an orally administered drug). The definition of concentration is amount divided by volume. We can rearrange the definition of concentration to find the amount of drug required to produce any desired concentration for a known volume:

[pic]

15

Many introductory pharmacokinetic texts suggest using this formula to calculate the “loading bolus” required to achieve a given concentration. This concept is often applied to theophylline and digitalis. The problem with applying this concept is that there are several volumes: V1 (central compartment), V2 and V3 (the peripheral compartments), and Vdss, the sum of the individual volumes. V1 is usually much smaller than Vdss, and so it is tempting to say that the loading dose should be something between,[pic]16 and [pic]17.

[pic]

Figure 20: Plasma drug concentrations following bolus doses based on target concentration times V1 and target concentration times Vdss.

As shown in figure 20, with multicompartment drugs administering a bolus of [pic]18will achieve the desired concentration for an initial instant, but the levels will rapidly decrease below the desired target. Administering a bolus of [pic]19will produce an overshoot in the plasma that may persist for many minutes. One resolution is to suggest that the dose be between these extremes.

Again, using an anesthetic drug as an example, consider the dose of fentanyl required to attenuate the hemodynamic response to intubation when combined with thiopental. The target concentration for this is approximately 3 μg/ml. The V1 and Vdss for fentanyl are 13 liters and 360 liters, respectively. The above equations can thus be interpreted as suggesting that an appropriate dose of fentanyl to attenuate the hemodynamic response is between 39 μg (3 ng/ml × 13 liters) and 1,080 μg (3 ng/ml × 360 liters) (figure 24). Personally, I don't need equations to tell me that the right fentanyl dose is somewhere between 39 and 1080 μg!

The usual dosing guidelines for bolus injection, as presented above, are oriented towards producing a specific plasma concentrations. Since the plasma is not the site of drug effect, it is illogical to base the calculation of the initial bolus on a plasma concentration. As pointed out previously, by knowing the ke0 of an intravenous anesthetic, we can design a dosing regimen to that yields the desired concentration at the site of drug effect. Returning again to figure 15, we can see the relative plasma and effect site concentrations following an IV bolus of fentanyl. The plasma concentration decreases continuously, while the effect site concentration rises until it reaches the plasma concentration, at which point both decrease continuously. If we do not want to overdose the patient, we should select the bolus that produces the desired peak concentration in the effect site.

The decline in plasma concentration between the initial concentration following the bolus (amount / V1) and the concentration at the time of peak effect can be thought of as a dilution of the bolus into a larger volume than the volume of the central compartment. This introduces the concept of Vdpeak effect, which is the volume of distribution at the time of peak effect. The size of this volume can be readily calculated from the observation that the plasma and effect site concentrations are the same at the time of peak effect:

[pic]

where Cplasma (peak effect) is the plasma concentration at the time of peak effect. Remembering that concentration is amount over volume, we can rearrange the above equation by substituting the initial plasma concentration times V1 for the loading dose. This gives the relationship:

[pic]

where Cplasma (initial) is the initial concentration following a bolus, Cplasma (peak effect) is the concentration at the time of peak effect, and the ratio of these is the percent decrease in plasma concentration between the initial concentration and the concentration at the time of peak effect.

Returning to the goal of selecting the dose to produce a certain given effect without producing an overdose: by definition the plasma concentration at the time of peak effect is the loading dose / Vd (peak effect). This can be rearranged to calculate the size of the initial bolus:

[pic]

The Vdpeak effect for fentanyl is 75 liters. To produce a peak fentanyl effect site concentration of 3.0 ng/ml requires 225 μg, which will produce a peak effect in 3.6 minutes. This is a clinically reasonable suggestion, compared with the absurd suggestion, based upon V1 and Vdss, of simply picking a dose between 39 and 1080 μg.

The exact same concept applies to oral dosage. Measure the plasma drug concentration at the time of oral dosage, and calculate Vdpeak effect as the oral dose divided by the concentration at the time of peak effect. The dose necessary to produce any desired concentration is then the target times Vdpeak effect.

Maintenance infusion rate

As previously pointed out, the rate at which drug exits from the body is the systemic clearance, Q, times the plasma concentration. To maintain a steady concentration, CT (for target concentration), drug must be delivered at the same rate that drug is exiting the body. Thus, the maintenance infusion rate is often presented as:

Maintenance infusion rate = CT × Q

For drugs with multicompartmental pharmacokinetics, which includes all of the drugs used in anesthetic practice, drug is distributed into the peripheral tissues as well as cleared from the body. The rate of distribution into tissues changes over time as the tissue concentrations equilibrates with the plasma. The above equation is only correct after the peripheral tissues have equilibrated with the plasma, which requires many hours. At all other times, this maintenance infusion rate will be too slow.

However, in some situations this simple maintenance rate calculation may be acceptable when combined with a bolus based on Vdpeak effect. For drugs with a long delay between the bolus dose and peak effect, much of the distribution of drug into the tissues may have occurred by the time the effect site concentration reaches a peak. In this case, the maintenance infusion rate calculated as CT × Q may be fairly accurate because Vdpeak effect was sufficiently higher then V1 to account for the much of the distribution of drug into peripheral tissues. This is the reason that the loading infusion-maintenance infusion concepts works modestly well for theophylline.

Another approach is the 2-step infusion rate proposed by Wagner.[?] In this method, two infusions, I1 and I2 are administered in sequence, with T the duration of I1. The second infusion, I2, is calculated as the CT times Q, exactly as noted above. T, the duration of the first infusion, is selected based on a combination of convenience and the degree of overshoot in plasma concentration will be clinically accepted during the loading portion. The rate of I1 is then,[pic]20 where β is the 0.693/terminal half-life.

Most drugs used in anesthesia have sufficiently rapid equilibration between the plasma and the effect site that the Vdpeak effect approach does not adequately address the distribution phase (i.e., distribution into peripheral tissues continues far longer than between the time required for plasma-effect site equilibration). Similarly, the Wagner scheme usually cannot balance the need for a rapid onset of anesthetic effect with accurate maintenance of the desired concentration.

This leads us to consider a more sophisticated approach in designing infusion rates to maintain target concentrations for drugs with multicompartment pharmacokinetics. Since the net flow of drug into peripheral tissues decreases over time, the infusion rate to maintain any desired concentration also decreases over time. If the initial bolus has been based on Vdpeak effect, no infusion need be administered until the effect site concentration peaks. Following the peak in effect site concentration, the equation to maintain the desired concentration is (unfortunately):

[pic]

[pic]

Figure 21: the infusion rate required to maintain a constant plasma drug concentration.

The infusion rate calculated by the above equation is initially rapid, and the rate decreases over time, as shown in figure 21. At equilibrium (t →∞) the infusion rate decreases to CT×V1×k10, with is the same as CT ×Q. Few anesthesiologists would choose to mentally solve such an equation during administration of an anesthetic. Additionally, the solution requires that the rate be continuously adjusted downwards, a hassle few would tolerate. Fortunately, there are simple techniques that can be used in place of such a solving complex expression.

[pic]

Figure 22: Dosing nomogram showing maintenance infusion rates for several popular anesthetic drugs.

Figure 22 is a nomogram in which the above equation has been solved, showing the infusion rates over time necessary to maintain any desired concentration of four popular intravenous anesthetic drugs: fentanyl, alfentanil, sufentanil, and propofol. This nomogram is complex, so we will review it in detail. The vertical axis represents the target concentration. The horizontal axis is the time, since the beginning of the infusion. Envision a horizontal line drawn from the target concentration (on the vertical axis) across the graph to the right edge. The infusion rate required to maintain the target is given by the diagonal line that most closely intersects with this imaginary horizontal line (desired target concentration) at the desired point in time. Each diagonal line is associated with a particular infusion rate.

For example, to maintain a fentanyl concentration of 1.5 ng/ml, the appropriate rates are 4.5 μg/kg/hr at 15 minutes, 3.6 μg/kg/hr at 30 minutes, 2.7 μg/kg/hr at 60 minutes, 2.1 μg/kg/hr at 120 minutes, and 1.5 μg/kg/hr at 180 minutes. Alternatively, you could select different times of rate adjustment, and read different infusion rates from the nomogram. Using nomograms such as this, one can determine the frequency and time points of the rate adjustment depending on clinical convenience and assessment of how accurately the intravenous anesthetic needs to be administered and titrated.

Another approach is to use a computer-controlled infusion pump to solve the polyexponential infusion equation. As previously mentioned, many programs do exactly this. STANPUMP is a DOS program that drives an infusion pump to maintain any desired plasma or effect site concentration. STANPUMP is used to study the pharmacokinetics and pharmacodynamics of the intravenous drugs used in the practice of anesthesia. These programs provide a new, and hopefully improved, method of titrating intravenous drugs. Several such devices are currently before the FDA, and it is likely that they will be introduced into clinical practice over the next decade. If you wish, you can download STANPUMP via anonymous FTP from pkpd.icon.palo-alto.med. in the directory STANPUMP.DIR.

References

* Here, I'm going to use "ln" to refer to the natural logarithm. Some programs use "log" for log in base 10, and others use "log" to refer to log in base e. For clarity, I'll keep with ln, which is unambiguously in base e.

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