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Pharmacokinetics and Pharmacodynamics

David J. Greenblatt, Jerold S. Harmatz, Lisa L. von Moltke, and Richard I. Shader

Pharmacokinetics is a discipline that uses mathematical models to describe and predict the time-course of drug concentrations in body fluids. During the last 30 years, our ability to apply pharmacokinetic principles to clinical psychiatry and psychopharmacology has undergone major advances. Important improvements in analytical chemistry techniques partly explain this progress. Methods for quantitation of drug concentrations in plasma and tissues are now highly sensitive and specific. For almost all drugs used to treat mental disorders, techniques such as gas chromatography, liquid chromatography, and mass spectroscopy can be used to quantitate serum or plasma concentrations of these agents and their pertinent metabolites after single therapeutic doses or during chronic therapy. These improved measurement methods have facilitated the application of pharmacokinetic models and have increased the role of therapeutic drug monitoring in clinical psychopharmacology. For some psychotropic drugs, ranges of therapeutic and potentially toxic plasma concentrations are reasonably well established; for others, such ranges are tentatively established or postulated, and research is ongoing. Clinicians are also increasingly familiar not only with how plasma drug concentrations can be helpful, but also with the potential pitfalls of therapeutic monitoring (11).

The discipline of pharmacokinetics has also been advanced by the general availability of iterative computer methods for nonlinear regression analysis. In the precomputer era, only the simplest modeling problems could be solved by approximation, usually after data transformation or linearization. Now, complex problems involving simultaneous determination of a number of clinically relevant pharmacokinetic variables can be rapidly solved, including numerical estimates of the statistical strength of the solution.

Pharmacodynamics, the study of the time course and intensity of drug effects on the organism, also has undergone major advances, largely due to technological advances in our capacity to measure drug effects. Kinetic–dynamic modeling uses mathematical methods to link drug concentrations directly to clinical effects. Iterative nonlinear regression procedures expeditiously solve the problems of fitting theoretical models to actual data (see Alzheimer's Disease: Treatment of Noncognitive Behavioral Abnormalities).

Approaches to Pharmacokinetic Modeling

Lipid-soluble psychotropic drugs generally do not behave as if the body were a single homogeneous space. The two-compartment model resolves the living organism into two distinct mathematical spaces, thereby enhancing the precision of describing and predicting drug behavior (Fig. 1). The assumptions of this model are as follows: (a) Intravenously administered medications are introduced directly into the "central" compartment, which consists of the circulating blood as well as other high-flow tissues such as brain, heart, lung, liver, and endocrine organs. (b) Irreversible drug elimination, either by hepatic biotransformation or renal excretion, takes place only from the central compartment. (c) Reversible distribution occurs between central and peripheral compartments, with a finite time (usually between 30 min and 6 hr after intravenous dosage) required for distribution equilibrium to be attained. (d) The peripheral compartment is usually inaccessible to direct measurement and is not a site of drug elimination or clearance (13, 18, 20, 43).

This model predicts that rapid intravenous drug administration will yield a profile of drug disappearance from serum or plasma that is consistent with a linear sum of exponential terms.

*C = Ae-*at + Be-bt [1]

In this equation, *C* is the plasma drug concentration at time *t*
after rapid intravenous injection. *A* and *B* are intercept terms
having units of concentration, and a and b
are exponents having units of reciprocal time. The exponents a
and b are related to, but are not equal to, the intrinsic
rate constants associated with the two-compartment model (Fig.
1).

A real clinical study typically yields a series of concentration-time data
points, which, when plotted on a logarithmic concentration axis, appears biphasic
(Fig. 2).
The investigator's task is to determine the parameters *A*, *B*, a,
and b from Eq. 1 which yield the best fit to the
data points. This is done by computer using iterative least-squares regression
methods (29). The function of best fit can then be used to calculate clinically
useful pharmacokinetic parameters (Fig.
3).

The biphasic profile of drug disappearance following rapid intravenous injection of lipid-soluble psychotropic drugs has important clinical implications. In physiological terms, the initial phase of rapid drug disappearance is attributable mainly to distribution, whereas the slower phase that follows is largely due to elimination or clearance. Distribution, elimination, and duration of action are interdependent. A drug's elimination half-life refers to the apparent half-life of disappearance in the post-distributive elimination phase. This half-life value does not necessarily correspond to its duration of action, which actually is determined by the relation of the actual plasma concentration to a minimum effective concentration. Drug action may be terminated during the distribution phase and have little to do with the value of the elimination half-life measured in the postdistributive phase (21, 22). The absolute size of the dose is also of critical importance, since a proportional change in dose will produce corresponding changes in all plasma concentrations, which in turn may disproportionately increase duration of action (Fig. 4). The same principles are also applicable following oral drug administration, although the mathematics becomes more complex.

Refinements in computerized techniques for nonlinear least-squares regression now allow distinct but interdependent sets of data points to be analyzed simultaneously. An example is the plasma drug concentration–time relationship during and after zero-order (constant-rate) intravenous infusion for a drug having a kinetic profile consistent with a two-compartment model. During the infusion, plasma concentrations (*C*) are related to time after the start of the infusion (*t*) as follows:

[2]

After the infusion, the relationship is

[3]

In these equations, a and b
again are exponents having units of reciprocal time, *T*inf
is the duration of the infusion, *Q* is the infusion rate, and *CL*
is the drug's metabolic clearance. The coefficients *X*1
and *X*2 are related to a,
b, and *K*E. Equation 2,
which represents an ascending function, is applicable during the time that the
infusion is ongoing (0 ≤ *t* ≤ *T*inf). The
infusion ends at *t = T*inf, at which time Eq. 3 becomes
applicable (*t* ≥ *T*inf); this equation represents
a descending function. Taken together, Eqs. 2 and 3 are continuous at all points
in time, but at *t = T*inf, the change point, the collective
function does not have a first derivative.

Physiologically, the pharmacokinetic properties of a particular drug should
be unique and constant for a particular individual subject, as long as no intervening
factors alter drug distribution or clearance in that individual. Under this
assumption, the same values of a, b,
*Ke*, and *CL* should be applicable both during
and after an intravenous infusion. Accordingly, Eq. 2 is fitted to concentration–time
points measured *during* the infusion, simultaneous with fitting Eq. 3
to points measured *after* the infusion (Fig.
5). The result is a single set of parameters (a,
b, *Ke*, and *CL*)
that are consistent with the drug's behavior both during and after the infusion.

The same approach can be extended to circumstances in which the same drug
is given to the same individual on different occasions. Fig.
6 shows actual, measured concentration–time points, along with
functions of best fit as consistent with Eqs. 2 and 3, in a three-way cross-over
study of the benzodiazepine derivative midazolam. A fixed total dose of midazolam
(0.1 mg/kg) was administered to a healthy volunteer on three separate occasions
by zero-order infusion. The rates of infusion differed across the three trials
as follows: 1 min, 1 hr, and 3 hr. Assuming that pharmacokinetic parameters
for midazolam in this particular subject are constant from trial to trial, Eqs.
2 and 3 can be fitted to all sets of data points simultaneously, with values
of *Q* and *T*inf appropriately adjusted to correspond
to the duration of infusion for each trial. The result is a single set of pharmacokinetic
parameters that is consistent with drug behavior during and after each of the
infusions across the three trials (Fig.
6).

Stable Isotopes in Clinical Pharmacokinetics

Techniques in synthetic chemistry allow the preparation of drug entities labeled with nonradioactive "heavy" isotopes of carbon or nitrogen, differing from the natural isotopes by as little as one molecular weight unit. If one carbon or nitrogen atom in a drug's customary structure is replaced by a heavy isotope of the same atom, the result is a nonradioactive stable-isotope-labeled (SIL) drug. The customary and SIL drug forms are identical, except that the SIL form differs by one atomic mass unit. These two forms cannot be distinguished using analytical techniques such as high-pressure liquid chromatography or gas chromatography, since the change in molecular weight by itself does not cause a measurable change in retention time. However, the customary drug form can be distinguished from the SIL form by gas chromatography (GC)/mass spectroscopy (MS), because one molecular weight unit will separate the corresponding fragments of the two drug forms (38).

Assuming that the customary and SIL drug forms have identical pharmacokinetic
properties in vivo, SIL methodology can substantially extend the power of clinical
pharmacokinetics (4, 5, 6). Bioequivalence studies traditionally require a cross-over
design, in which the same drug is administered to the same individual on separate
occasions. One trial involves dosage with the "reference" compound; in the other
trial, the dosage form being tested is administered. Plasma drug concentrations
are measured at multiple time points after drug administration in each trial,
and bioequivalence is evaluated by comparison of variables such as peak plasma
concentration (*C*max), time of peak concentration
(*T*max), elimination half-life, and area under the
plasma concentration curve (AUC). A similar crossover design is used to compare
the rate and completeness of drug absorption after oral or intramuscular administration
relative to intravenous dosage. With SIL methodology, bioequivalence studies
can be done with a single exposure trial per subject. One dosage form is the
conventional drug; the other is the SIL form. Both are administered at the same
time. The GC/MS analytical procedure can simultaneously quantitate concentrations
of each drug form in each plasma sample. This approach saves time and reduces
drug exposure and blood sampling requirements for human volunteers. It also
reduces variance due to sequence effects. After oral administration, for example,
*C*max and *T*max may vary
within the same individual when the same dosage form of the same drug is administered
on different occasions. AUC may also vary from time to time due to subtle changes
in clearance, regardless of dosage form or rate of administration.

Stable-isotope-labeled methodology can also be used to study definitively the single-dose pharmacokinetics of a drug in a patient already receiving the same drug on a chronic basis. Under conventional circumstances, the dosing rate divided by the steady-state plasma concentration yields an estimate of clearance, assuming complete absorption of each oral dose. However, elimination half-life and volume of distribution cannot be determined unless the drug is discontinued. A single intravenous dose of SIL drug can be used to determine all pharmacokinetic parameters without interfering with ongoing therapy and with no assumptions about drug absorption.

SIL methodology has drawbacks as well as benefits. Substantial synthetic and medicinal chemistry resources are needed to prepare SIL drugs and formulate them for human administration. GC/MS facilities must be available for analysis of all biological samples. These analyses are much more difficult and expensive than usual high-performance liquid chromatographic and GC procedures.

Mathematical Modeling of Drug Interactions In Vitro

The serotonin-specific reuptake inhibitor (SSRI) antidepressants, of which fluoxetine is the prototype, are now prescribed extensively in clinical practice. The class of SSRIs has the secondary pharmacologic property of reversibly inhibiting the activity of hepatic drugmetabolizing enzymes. Following the introduction of fluoxetine, experimental and clinical reports of drug interactions began to appear, in which coadministration of fluoxetine caused decreased clearance and elevated plasma concentrations of certain other coadministered medications (8). Some of these interactions have been of clinical importance. The various SSRIs (and their metabolites) that are now available, and those under investigation, are not equivalent in their capacity to inhibit drug metabolism, nor is the metabolism of every drug equally affected by SSRI coadministration.

Literally hundreds of clinical pharmacokinetic studies would be needed to delineate which drugs will interact with SSRIs, the extent of the interactions, and whether they can be safely coadministered under usual clinical circumstances. However, recent advances in the field of cytochrome chemistry have made it possible to study drug interactions using in vitro models, thereby making at least some clinical pharmacokinetic studies unnecessary (2, 3, 14, 15, 26, 27, 42). Using ultracentrifugation techniques, the microsomal drug-metabolizing component of human liver tissue can be isolated (41). When mixed with appropriate reaction cofactors, human liver microsomes will metabolize drugs in vitro much as happens in vivo.

Nonlinear least-squares regression techniques are applied to enzyme kinetics in much the same way as in clinical pharmacokinetics. In a typical in vitro enzyme kinetic study, varying concentrations of a drug substrate (*S*) are incubated with human liver microsomes for a fixed period of time. The amount of metabolic product generated is quantitated in the reaction mixture, and converted to a normalized reaction velocity (*V*), usually in units of nanomoles of product per minute per milligram of microsomal protein. In most cases the relation of *V* to *S* is consistent with the following equation:

[4]

where *V*max is the maximum reaction velocity and *Km* is the substrate concentration at which *V* is 50% of *V*max. Linearizing transformations of data, with their inherent distorting effects, were commonly used in the precomputer era to estimate values of *V*max and *Km* from experimental data. However, current computerized methods allow direct analysis of untransformed data by nonlinear regression. Reliable analysis of in vitro metabolic inhibition studies likewise is dependent on nonlinear regression methods. If a metabolic inhibitor is added to a reaction mixture, the reaction velocity will be depressed depending on the concentration of substrate, the concentration of inhibitor (*I*), and an inhibition constant (*K*i) which reflects the inhibiting "potency" of the inhibiting compound. Numerical values of *K*i reciprocally reflect inhibiting potency; that is, low values of *K*i indicate high inhibiting potency.

If competitive inhibition is the mechanism of the inhibiting effect, the variables described above are related as follows:

*V* = (*V*max · *S) /
S + Km* · (1 + *I*/*K*i)
[5]

A typical in vitro study will measure *V* at varying concentrations of
*S* and *I*. *V*max, *Km*,
and *K*i can be determined using linearizing transformations
of data, but these have the potential to distort the relationships (1). Computerized
nonlinear regression can simultaneously analyze all data points without transformation,
and provide best-fit estimates of *V*max, *Km*,
and *K*i.

In vitro drug interaction studies have provided data directly useful to
clinical decision making. Coadministration of the SSRI fluoxetine with the cyclic
antidepressant desipramine (DMI) impairs metabolic clearance of DMI and significantly
elevates steady-state plasma levels of DMI in humans (35). The biotransformation
of DMI involves hydroxylation by the hepatic microsomal system, yielding 2-hydroxy-desipramine
(2-OH-DMI). This reaction is mediated by Cytochrome P450-2D6, of which fluoxetine
and its principal metabolite, norfluoxetine, are competitive inhibitors. In
vitro studies using human liver microsomes, based on methods described above,
provide *K*i estimates for fluoxetine and norfluoxetine
with regard to their capacity to inhibit formation of 2-OH-DMI from DMI (40)
(Fig. 7).
Based on steady-state plasma concentrations of fluoxetine and norfluoxetine
measured in humans, the expected partitioning of these two drugs between plasma
and liver, and *Ki* values measured in vitro, impairment
of DMI clearance in vitro during coadministration of fluoxetine can be qualitatively
and quantitatively predicted (40). Similar methods correctly predict that the
SSRI paroxetine also will strongly impair DMI clearance in vivo. In contrast,
the SSRI sertraline, its metabolite desmethylsertraline, and the SSRI fluvoxamine
are weak inhibitors of DMI clearance in vivo (35), and weak inhibitors of 2-OH-DMI
formation from DMI in vitro (40).

After a single dose of a psychotropic medication, its plasma concentrations and clinical effects will increase, reach a maximum, and then decline with time. Pharmacokinetics deals with the time-course of drug concentration, whereas the discipline of pharmacodynamics refers to the time-course and intensity of drug action or response. The capacity to understand and predict individual differences in drug response is of critical importance in psychopharmacology, since the objective of drug treatment is to produce therapeutic benefit while minimizing side-effects. Advances in the clinical applications of pharmacodynamics are due to improved precision and objectivity in methods for measuring human drug response, and the emergence of the discipline of kinetic–dynamic modeling, in which drug concentration is used as a direct predictor of response.

Methods of Measurement

Each class of psychotropic drugs produces unique pharmacodynamic effects, as well as problems with measurement. Pharmacodynamic studies of sedative–anxiolytic drugs serve to illustrate available approaches to quantitation, and the limitations associated with each (24, 31).

Benzodiazepine derivatives, and other g-aminobutyric acid (GABA)-benzodiazepine receptor agonists, produce clinical sedation as a final common pathway of action (16, 28). However, the clinical consequences of benzodiazepine agonist action depend on the therapeutic objective and the setting of drug administration. Antianxiety– antipanic effects, enhancement of sleep, and reduction of seizure activity are usually described as primary therapeutic actions. Typical side-effects include excessive or persistent sedation, drowsiness, ataxia, incoordination, slowed reaction time, or slowed psychomotor performance. Some clinical actions are ambiguous; impaired memory, for example, may be desirable in the context of premedication before general anesthesia or sedation prior to endoscopy or cardioversion, but undesirable in other circumstances. Benzodiazepines also produce some effects whose direct link to clinical efficacy or side-effects are not directly established. These include: increased beta activity on the electroencephalogram (EEG), reduced saccadic eye movement velocity, increased postural sway, elevated concentrations of plasma growth hormone, and reduced plasma concentrations of cortisol and ACTH.

Measurement of the pharmacodynamic effects of benzodiazepine agonists becomes more reliable and objective as the effects become more removed from the primary therapeutic action. The time-course and intensity of primary therapeutic effects are the most difficult to measure. Clinical anxiolytic or sedative properties can be quantitated using global assessments by patients or trained observers, or by rating instruments targeted to specific symptoms or symptom groups. These measurement approaches have been extremely useful in clinical psychopharmacology, but have limitations. They are subjective ratings, and can be influenced by the outlook, expectation, and experience of the patient as well as the observer. Quantitative drug effects must be evaluated as change scores relative to both the pretreatment baseline condition and, to inactive placebo. If special populations (such as the elderly) have a unique way of interpreting rating scale items or describing drug effects, this further complicates the methodology.

Procedures quantitating secondary effects of benzodiazepine agonists generally focus on aspects of psychomotor performance, such as measures of reaction time, speed, and accuracy of task performance, and the capacity for information acquisition and recall. These measures are only partly objective, although they provide numeric results. Again, change scores are the most meaningful; drug effects must be compared to pre-dose baseline performance, and to effects of placebo. Psychomotor performance procedures are influenced by practice—subjects' performance will progressively improve if they repeatedly perform a task in the absence of medication. This can complicate interpretation of drug-associated changes, and must be accounted for in the experimental design. The study of special populations (such as the elderly) is also affected by baseline performances that differ from that of the control group. It is not clear how best to incorporate differences in baseline performance when comparing drug effects (25). The relation of drug effects on laboratory performance tasks to effects on real life tasks is not clearly established. Even for complex and seemingly realistic laboratory tasks such as simulated automobile operation, the consequences of error are negligible in the laboratory, but potentially lethal on the road.

Fully objective measures of pharmacodynamic effects of benzodiazepines are of increasing interest. Computerized analysis of the EEG can directly quantitate benzodiazepine effect, either by evaluation of activity in the beta frequency band by fast-Fourier transform, or by aperiodic analysis (12, 21, 22, 23, 32, 37, 39). In addition to being completely objective, these measures are unresponsive to placebo, and are not altered by practice or experience. No theoretical basis is yet established to link EEG changes to changes in therapeutic effects or psychomotor side effects. Empirical data, however, demonstrate that the pharmacodynamic profile of EEG changes following benzodiazepine agonist administration closely intercorrelates with other measures of drug activity (12, 23).

Kinetic–Dynamic Modeling

Kinetic–dynamic modeling procedures directly evaluate the relation of drug
concentrations (*C*) to drug effect (*E*), with the objective of determining
how much variability in drug effect is attributable to measurable drug concentrations
(9, 10, 12, 30, 32, 36, 37). The Sigmoid-*E*max relationship
is commonly used in kinetic–dynamic modeling. When this model fits the data,
investigators are reassured about the concentration–effect relationship, since
many drug-receptor interactions fit the same model. The linkage is described
by the following equation:

[6]

*E*max is the maximum drug effect (relative to baseline),
which cannot be exceeded no matter how high the drug concentration; EC50,
is the concentration at which the drug effect is 50% of *E*max;
*A* is an exponent of unknown biological significance related to the "steepness"
of the curve in its linear phase (Fig.
8). When applicable, this model allows quantitation of individual
differences in sensitivity to a given drug, relative potencies of drugs having
the same mechanism of action, and the quantitative and qualitative effects of
pharmacologic potentiators or antagonists. However, not all data sets are consistent
with the Sigmoid-*E*max model, and it may be inappropriate
or even misleading to "force" this model on data which it does not fit (17).
In many cases, data from kinetic– dynamic studies can be described by an exponential
function such as:

*E = B · CA * [7]

or by a linear function such as:

*E = m · C + b* [8]

Plasma Concentration Versus Effect-Site Concentration

The use of plasma drug concentration as the independent variable in kinetic–dynamic
modeling procedures (Eqs. 6, 7, and 8) assumes that equilibration of drug between
plasma and the site of action (effect-site) in brain is very rapid, such that
plasma concentrations are proportional to concentrations at the effect site.
This assumption may need modification in some circumstances. For the subject
described in Fig. 2
and Fig. 3,
maximum change over baseline in EEG beta activity is delayed following completion
of the infusion of midazolam (Fig.
9). A kinetic–dynamic plot of EEG effect versus plasma concentration
reveals "counterclockwise hysteresis," in which the maximum effect does not
correspond to the maximum plasma concentration. This may be explained by the
time necessary for midazolam to equilibrate between plasma and effect site (19).
To account for this delay, the model is modified to include the hypothetical
effect site (Fig.
10). Effect-site concentrations, as opposed to plasma concentrations,
are presumed to determine pharmacodynamic effects (9, 36, 37). The first-order
rate constant *K*EO governs the exit of drug from the
hypothetical effect site, and also determines the observed equilibration of
drug between plasma and effect-site. *K*EO can be used
to calculate a half-life of equilibration (*t*1/2KEO).
If *K*EO is incorporated as a variable in the iterative
analysis, the resulting plot of effect-site concentration versus EEG effect
eliminates the hysteresis and yields data points that are consistent with Eq.
7 (Fig. 11).

The major challenge for health care professionals involved in clinical psychopharmacology is to understand and compensate for individual variations in drug response. Why does a given dose of a given psychotropic medication produce different clinical responses among individuals comprising a population of patients? When the same dose produces different plasma concentrations in different individuals, the variance is pharmacokinetic. When the same concentration produces different responses in different individuals, the variance is pharmacodynamic. The principles described in this chapter, as they become increasingly available to and used by clinicians, should enhance our ability to deliver effective drug treatment with minimal side effects.

This work was supported in part by grant MH-34223 from the Department of Health and Human Services. Dr. von Moltke is the recipient of an Abbot Laboratories Fellowship in Clinical Pharmacology.

The authors are grateful for the collaboration and assistance of Drs. Bruce L. Ehrenberg and Lawrence G. Miller.

*published 2000*