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```					December 2, 1995                                                                           Page 1

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 pharmaco-
kinetics. 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.
December 2, 1995                                                                           Page 2

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.
December 2, 1995                                                                                    Page 3

The physiologic basis of pharmacokinetics and pharmacodynamics

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          Figure 1: Volume of distribution represents dilution of
insight. Let's assume the size of the bucket is drug into a volume
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.

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            Figure 2: Clearance represents the flow of blood or
clearance does not depend on the concentration of      plasma cleared of drug
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.
December 2, 1995                                                                                            Page 4

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
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
Figure 3: The one compartment model, with a single volume
compartment model). The volume to the left        and flow term.
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.”

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                Figure 4: Two compartment pharmacokinetic model, with
two volumes, (central and peripheral) and two clearances
partitioning of drug into all body structures at     (central, and 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
Figure 5: Three compartment model, with three volumes,
some insight, particularly for highly lipophilic
(central, rapidly equilibrating peripheral and slowly
drugs in which a large V3 may be explained by         equilibrating peripheral) and three clearances (central, rapid
extensive distribution of the drug into fat. As       and slow intercompartmental).
we will see later on, the volumes and
December 2, 1995                                                                              Page 5

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.1,2 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.3 Lastly, drugs themselves can alter hepatic
blood flow. For example, halothane (an inhalational anesthetic) decreases liver blood flow by
60% in dogs.4,5

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,6 as
is creatinine clearance, which can be predicted from age and weight:7
December 2, 1995                                                                               Page 6

Men:
(140 - age ) x weight( kg )
Creatinine Clearance ( ml / min ) =
72 x serum creatinine ( mg %)
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.8 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.9

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 pathology10
although some studies have identified a small decrease associated with age.11 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

Decreases in cardiac output would be expected to decrease intercompartmental clearance.
The net effect of decreased intercompartmental clearance is to increase the plasma concentrations
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.
December 2, 1995                                                                                         Page 7

Central volume of distribution

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 concen-
tration has peaked at time 0. Instead, the concentra-
Figure 6: True and plasma concentrations following
tion of drug in the arterial blood immediately after
bolus injection.
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.

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

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
December 2, 1995                                                                               Page 8

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:
X total drug
C plasma =
V total body

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

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.20 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).21 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 trazodone22 and nitrazepam.23 However, as mentioned before, the partitioning of drugs into
December 2, 1995                                                                               Page 9

body tissues is dictated by physicochemical properties of the drug itself, and thus the partitioning,
per unit of tissue, is not influenced by age.24

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
December 2, 1995                                                                              Page 10

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,25,26 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.27

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,28
mepivacaine,29 and prilocaine.30 The enantiomers of ketamine also have been extensively
studied,31,32 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,33 the difficulty of doing such
studies has precluded more widespread analysis.
December 2, 1995                                                                               Page 11

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
December 2, 1995                                                                               Page 12

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          Figure 8: x = x0 e-kt
sides of the above equation, we get:

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

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

x0
x   ln( x0 ) - ln( )
k =    =               2
t        t1 / 2

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

*
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.
December 2, 1995                                                                                  Page 13

We can simplify the numerator to:
        
x0       x0     
ln( x0 ) - ln( )  ln         
2        x0     
 2      

 ln(2)
 0.693
This neatly relates the slope, k, to the time required for a 50% change, t ½:
0.693
k =
t1/ 2
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:
0.693
t1/ 2 =
k
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.

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:
Figure 10: The one-compartment model.
dx
(rate) = Q x C
dt

This is a first-order process. We can find k, the rate constant, by substituting x/V for C in
the above equation:
December 2, 1995                                                                              Page 14

dx
( rate)  QC
dt
x
Q
V
Q
    x
V
 k x (definition of a first order process, page 11)

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:
              
x0  - Q t    x V    x
AUC =  A e-kt dt =            e V  dt = 0   0
      
0              0
V              V Q    Q

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
December 2, 1995                                                                              Page 15

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:
dx
= I - kx(t)
dt

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:
I
x(t) =     (1 - e-kt )                      Equation 1
k
As t , , e-kt  0, and the above equation reduces to:
I
x(  ) =                                    Equation 2
k
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:
I   I
=   (1 - e-kt )
2K   K
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.
December 2, 1995                                                                            Page 16

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:
x(  )      IV       I
C(  ) =           =         =
V        QV        Q
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
December 2, 1995                                                                           Page 17

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
x0 -kt
C(t) =        e
V
3)     If you know two separate concentrations, C(t1) and C(t2), you can calculate k as:

ln(C( t 1 )) - ln(C( t 2 ))
k =
t 2 - t1

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):
December 2, 1995                                                                           Page 18

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 g/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 g/ml  20 liters = 40 mg.

3)     What cephprololopam infusion is needed to maintain a cephprololopam concentration of
2 g/ml?
Answer: I = Ctarget Q (clearance) = 2 g/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
g/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:
systemic dose
F
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:
I(t) = F Doral k a e-k a t
December 2, 1995                                                                                       Page 19

where F is the fraction bioavailable, Doral is the dose given oral, and ka is the absorption rate
constant. Note that the integral of ka e  ka t 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:
dx
= I(t) - k x = F D k a e-k a t - k x
dt

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:
D F k a -k a t
x(t) =            (e     - e-kt )
k - ka

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.

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         Figure 11: Shape of curve following intravenous bolus
linear.                                               injection.

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
December 2, 1995                                                                                       Page 20

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:
C(t) = A e-t + B e- t + C e-t                                 Equation 3
where t is the time since the bolus, C(t) is the drug concentration following a bolus dose, and A,
, B, , C, and  are parameters of a pharmacokinetic model. A, B, and C are called coefficients,
while , , and  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/ + B/ + C/).

Equation 3 really says that the
concentrations over time are the algebraic sum
of three separate functions, Ae-t, Be-t, and Ce-
t
. 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:
Cp(t) = A + B + C

The sum of the coefficients A, B, and
C, equals the concentration immediately              Figure 12: Figure 11 as sums of exponentials
December 2, 1995                                                                               Page 21

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  component, because Ae-
t
>> Be-t >> Ce-t.

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  >  > . However,
for historical reasons some individuals always call the smallest exponent . It is usually clear
from the context which exponent is the smallest. Instead of calling them , , and , some
individuals refer to them as 1, 2, and 3.

There is a special significance to the smallest exponent. Let  represent the exponents, ,
, and . As t  , t  , and e-t  0. Now, as t  , t will go to  the most slowly for the
smallest value of , and hence e-t will go to 0 the most slowly for the smallest value of  (i.e., 
as notated in equation 3). After enough time has passed (approximately t > (log(C) - log(B))/(-
), if you're curious), the values of Ae-t and Be-t are so close to 0, relative to the value of Ce-t
that the drug concentrations over time are pretty much following the concentrations predicted just
by Ce-t, which will appear to be a straight line, with a slope of , 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/ and 0.693/, and a terminal half-life (sometimes called the “elimination
half-life”), calculated as 0.693/. 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
Varvel34 and Hughes et al,35 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
December 2, 1995                                                                                              Page 22

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).
dx1
= I + x 2 k 21 + x 3 k 31 - x1 k 10 - x1 k 12 - x1 k 13 = I + x 2 k 21 + x 3 k 31 - x1 ( k 10 + k 12 + k 13 )
dt
dx 2
= x1 k 12 - x 2 k 21
dt
dx 3
= x1 k 13 - x 3 k 31
dt
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 x, and dt becomes t.
With a t 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.
December 2, 1995                                                                                         Page 23

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.

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       Figure 13: Context sensitive half-times (vertical axis)
for these drugs are 2 hours and 9 hours,                  for sufentanil and alfentanil, as a function of infusion
respectively. Even though sufentanil has terminal         duration (horizontal axis).
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.
December 2, 1995                                                                                       Page 24

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: The times for a 20%, 50%, and 80%
Figure 14 shows the times required for the              decrease in sufentanil concentration, as a function of
sufentanil concentration to fall by 20, 50, and 80%     infusion duration.
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

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          Figure 15: The plasma (solid) and biophase
concentrations (dashed lines) following a
drug concentration in the microscopic environment of the
bolus of 3 common opioids.
December 2, 1995                                                                                       Page 25

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.

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
such small amounts of drug from the central
compartment that it has no influence on the
plasma pharmacokinetics.
Figure 16: The compartmental model, now with an added
effect site. ke0 is often directed outside, as though drug were
If a constant plasma concentration is     eliminated from the effect site.
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.36,37,38,39,40,41,42
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,43 and slow for morphine and ketorolac.
December 2, 1995                                                                                       Page 26

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),44 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
Figure 17: The plasma and effect site concentrations for
(descending) plasma concentration. The            propofol, assuming a t ½ ke0 of 1, 2.8 (the real value) and 5
plasma concentration continues to fall, and       minutes.
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.
December 2, 1995                                                                                    Page 27

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.

Let us now integrate our the sigmoidal
relationship between concentration and effect, the
concept of the therapeutic window, and the              Figure 18: The plasma and effect site concentrations
equilibration delay between the plasma and the site following a bolus or infusion of propofol.
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               Figure 19: The effect site concentrations over time for
concentrations after two different bolus doses of doses of fentanyl that target the high and low edges of the
fentanyl designed to achieve the high and low       therapeutic window.
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:
December 2, 1995                                                                                   Page 28

Amount = Concentration  Volume
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, Concentration  V 1 16 and
Concentration  Vd ss 17.

As shown in figure 20, with
multicompartment drugs administering a bolus of
Concentration  V 1 18will achieve the desired
concentration for an initial instant, but the levels
will rapidly decrease below the desired target.
Concentration  Vd ss 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.                                 Figure 20: Plasma drug concentrations following
bolus doses based on target concentration times V1
Again, using an anesthetic drug as an          and target concentration times Vdss.
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
g/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 g (3 ng/ml  13 liters) and 1,080 g (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 g!

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
December 2, 1995                                                                             Page 29

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:
Vd peak effect =
Cplasma (peak effect)
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:
C plasma (initial)           V1
Vd peak effect = V 1                        =
C plasma (peak effect)   percent decrease

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:

The Vdpeak effect for fentanyl is 75 liters. To produce a peak fentanyl effect site
concentration of 3.0 ng/ml requires 225 g, 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 g.

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
December 2, 1995                                                                                  Page 30

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

Another approach is the 2-step infusion rate proposed by Wagner.45 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
I2
then,             20 where  is the 0.693/terminal half-life.
1 - e - T

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):
maintenance infusion rate (t) = CT  V 1 ( k 10 + k 12 e-k 21t + k 13 e-k 31t )

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    Figure 21: the infusion rate required to
be used in place of such a solving complex expression.         maintain a constant plasma drug concentration.
December 2, 1995                                                                                 Page 31

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
g/kg/hr at 15 minutes, 3.6 g/kg/hr at 30 minutes, 2.7
g/kg/hr at 60 minutes, 2.1 g/kg/hr at 120 minutes, and Figure 22: Dosing nomogram showing
1.5 g/kg/hr at 180 minutes. Alternatively, you could      maintenance infusion rates for several popular
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
from pkpd.icon.palo-alto.med.va.gov in the directory STANPUMP.DIR.
December 2, 1995                                                                         Page 32

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