In its most basic
presentation, pharmacokinetic (PK) and pharmacodynamic (PD) data is simply the
two dimensional relationship between either drug concentration and time or
effect and drug concentration, respectively.
The purpose of PK/PD modeling is to 1) to summarize this data with
simple statistics or functions with a relatively small number of parameters and
2) to use statistical techniques to distinguish random variability within and
among subjects from fixed, structural aspects of drug disposition or
effect. The tools available for this
modeling are various techniques of statistical analysis, usually (but not
always) embodied in specific software packages.
Pharmacokinetics
To fully characterize drug
disposition, PK models are, if not
necessary, very helpful. It is
essential to formulate both “structural” and “error” models. Structural models characterize the
deterministic, or fixed, aspects of drug disposition. The best known and most
commonly used structural models are compartment models assuming instantaneous
mixing to simplify the mathematical
details. However, the assumption of instantaneous mixing is clearly incorrect
and standard compartment models tell us nothing about the early phases of drug
distribution. An exciting recent
advance is the development of recirculatory models, which can accurately
predict this mixing phase. These types
of models are particularly suited for understanding how hemodynamic variables
may impact on PK behavior.
Given
a specific structural model,
determination of PK parameters requires the assumption of an error
model. Whatever the structural model, the concentrations predicted by the model
and the measured concentrations will not agree. It is common to assume that the relationship between measured and
predicted (by the structural model) concentrations are related by the following
equation
Cm = Cp + M
where Cm is the measured concentration, Cp is the
predicted concentration and is a function of the structural PK parameters, and
ε is the “error” term. The term
“error model” refers to the assumed statistical distribution of ε. For example it is common to assume that
ε has a normal distribution. One then obtains estimates of PK parameters using maximum likelihood. The principle of maximum likelihood states
that the “best” estimate of parameters are those which maximize the likelihood
of the observations. If ε has a
normal distribution then the probability of any single observation is
Pi
= (1/√(2πvar) )EXP(-(Cm,i-Cp,i)2/(2 var))
where var is the variance of the error term (and can
be a function of the structural parameters).
The probability of a series of observations is equal to P= Πi
Pi. Maximizing the likelihood of the
observations is equivalent to maximizing the natural logarithm of
likelihood. In this case the logarithm
of P is equal to
log
likelihood = log P = Σi Pi = -nlog(√(2πvar)
– Σi (Cm,i-Cp,i)2/(2var)
By specifying the error model, i.e., by assuming a
model for var, and then maximizing the log likelihood with respective to the
parameters, consistent estimates of the same can be obtained. In the most basic application of this
method, it is assume that var is a constant and the equations reduce to simple
least squares. However, the assumption of a constant variance for the model
error is often unacceptable and a more realistic error model must be used. In this case the approach is known as
extended least squares. When the error model is relatively straightforward this
method is also very useful for the analysis of “dense” data, i.e., when there
is sufficient data from each subject for determination of PK parameters for the individual patient.
The most prominent focus of modern PK is
population analysis in which data from multiple individuals is pooled for
analysis. . The best known software for population analysis is
NONMEM, although Pharsight Inc also now offers software for population
analysis. The underlying mathematics is challenging since it requires
distinguishing inter-patient from intra-patient variability. But population
analysis offers great advantages, allowing the investigator to use sparse data
(few data points per individual), and
to assess inter-patient variability.
The mathematical challenge can be appreciated by considering
likelihoods. For any individual, the
probability of the observed results is a function of that individual’s
structural parameters (denoted S), i.e., Pi = f(S). However, when we pool data from multiple
patients we have to also consider that the values of S are different for
different patients. To calculate the likelihood we have to multiply the above
probability of the observed results for an individual given particular parameter values by the probability of those
particular values (denoted h(S)) and then sum (integrate) over all possible
values of S. In mathematical terms
Li =
∫ Pi (S) h (S) dS
And the likelihood of the all observations is
L =
∏i Li
In general, the integrals involved in the calculation
of the likelihood are intractable and simplifying assumptions are
required. It is imperative that the
investigator understands the nature of these assumptions.
The
mainstay of PD modeling is the Hill, or sigmoid Emax, equation, which
postulates the following relationship between drug concentration (C) and drug
effect (E)
E =
Emax (Cγ/(Cγ+C50γ))
where Emax is the maximal effect, C50 is the drug
concentration that results 50% of the maximal effect, and γ is the slope
parameter that determines the slope of the concentration-response curve. When
the effect is a continuous variable, estimates of Emax, C50 and γ are
usually obtained by extended least squares or iteratively reweighted least
squares when there is sufficient data for analysis of individual subjects. When sparse data is pooled from multiple
patients, then a population analysis is a better approach. However, the more interesting situation is
dichotomous data. For example, in the
analysis of anesthetic effect we often have binary data, the patient is either
responsive or nonresponsive. In this
situation the analysis is more complicated. The drug effect is the probability
of response or nonresponse. Emax by
definition is unity. C50 and γ are
typically estimated by maximum likelihood.
The likelihood of the observed results is
L =
Πi (PiRi)(1-Pi)1-Ri
where Ri is one if there is a positive anesthetic effect
and zero if there is not, and
Pi =
Ciγ/(Ciγ + C50γ)
and Ci is the concentration at the time of the
observation. At this time it is unclear
how reliable the analytic tools when there is sparse data (minimal number of
data points per patient). Simulations
reveal that C50 may be accurately estimated either by naïve pooling of data or
population analysis. However estimates
of γ may be significantly biased.