|
|
Terms
and Definitions
Variable
Variables
are any characteristic that are measured, controlled or manipulated
1
(e.g.: Concentration, time).
Dependent
variable
A
variable which is measured or registered and is expected to be
dependent on manipulation or experimental conditions1
(e.g.:Concentration in PK)
Independent
variable
A
variable which is manipulated or which is a predictor variable1
(e.g: Demographics)
Parameter
A parameter
is a model coefficient relating dependent and independent variables.
In PKPD these parameters relates to some mechanistic property of either
the drug or biology. (e.g.: Mean population Clearance(CL),between
subject variability of CL, CL in an individual).
Model
A model
is a quantitative representation of the relationships among the entities
in a system, often used to make predictions about the system. 2
In the
context of PKPD, a model is a mathematical form specifying the probability
distribution of random variables representing observations (here drug
concentrations and effect) in a subject-matter domain, and which may
be the composition of several submodels (e.g.: one for PK, another
for PD), including ones for the distribution of unobservable conceptual
entities (e.g.: drug clearance in the population)3
back
to top
Linear
and Non-linear model
A Linear
model can be function of the form
y
=ƒƒ(x;β) = β0+β1•x1+
β2•x2+…
where
x denotes the vector of independent variables
β denotes the vector of model parameters.
and in
which
- each independent
variable is multiplied by an unknown parameter,
- there is at most one
unknown parameter with no corresponding independent variable,
and
- all of the individual
terms are summed to produce the final function value.
A Nonlinear
model can be a function of the form
y
= ƒ(x;β) = β0+β0•
β1•x1
in which
- the functional part
of the model is not linear with respect to the unknown parameters
β0,β1…
One
way to determine if a model is linear or nonlinear in the model parameters
is to compute partial derivatives with respect to each of the parameters
in the model.
The
model is linear if none of the partial derivatives involve
model parameters
The model is nonlinear if the partial derivative with respect
to one parameter involve another parameter.
Fixed
effect parameter
Fixed effect
parameters are population parameters, which define the average value
for a parameter in a population and/or the average relationship between
measurable patient factors (creatinine clearance) and pharmacokinetic
parameters (renal clearance).4
Typical
value of a parameter is the average parameter value at specific covariate
value. (e.g.: 72L/hr/70kg). The typical value equals the average population
value when no covariates are used.
Random
effect parameters
Random
effect parameters are population parameters, which quantify random
(unknown) variation. Random effects can be subdivided into:
- Between subject variability
(BSV) - A measure of unexplained random differences between individuals.
BSV is also called as inter-individual variability.
- Residual variability
- A measure of remaining unexplained variability when all other sources of variability has been accounted for. It includes measurement error and model specification errors. Generally, statistical literature refers to this as within subject
variability (WSV) or intra-individual variability.
- Between occasion variability
(BOV) - A measure of unexplained random differences in an individual
between different occasions.
back
to top
Mixed
effects modeling
Population
analysis (PK or PD) is usually performed by the mixed effects modeling
approach. A mixed effect model mixes the effects of random quantities
(BSV,WSV) and fixed effects (age, weight, creatinine clearance) on
the observations i.e., plasma concentrations, allowing these influences
to be simultaneously quantified.4
Nonlinear
mixed effects modeling involves approximation during estimation. Linear
mixed effects modeling does not suffer from these approximations.
Linear models can be identified using the partial
derivative method and linear mixed effects modeling can be used
when needed.
Structural
model
The structural
model is the deterministic part of a population model. A complete
population PK / PD model usually constitutes of four structural and
three statistical (error) models. The four structural models include:
(1) PK model, (2), disease progression model, (3) PD model and (4)
covariate (or prognostic factor) model. The parameters of these models
are called as 'fixed effects'. Examples of fixed effects include the
typical value of systemic clearance in a 70 kg person and the mean
potency of the drug. (e.g.: a two - compartment model or an Emax
model).
Random
effect models are between subject variability (BSV) model, residual variability model and between occasion variability (BOV)
model.
1. Between subject
variability (BSV) model
BSV
refers to the random variability between individuals. Individual
kinetic parameters are modeled as functions of population means
and individual random deviations (η). η's are assumed
to be identically, independently distributed with N(0, ω2).
2. Residual
variability model
A residual
variability model includes intra-individual error, measurement
error and any model misspecification error and represents the
uncertainty in the relationship between the plasma concentrations
predicted by the model and the observed concentration.
3. Between occasion
variability (BOV) model
BOV
includes the variability in the pharmacokinetic parameters within
an individual between occasions. BOV is modeled by lumping it
in the residual variability in certain softwares (e.g.: NONMEM)
Population
Pharmacokinetics and Pharmacodynamics
Population
pharmacokinetics is the study of the sources and correlates of variability
in drug concentrations and effects among individuals who are the target
population receiving clinically relevant doses of a drug of interest.(As
adapted from FDA
guidance)5
Mostly
the phrase population analysis is being used to refer to one-stage
analysis when it is equally applicable to standard two stage approach.
Maximum
likelihood approach
A
maximum likelihood approach finds that estimate of a parameter, which
maximizes the probability of observing the data given a specific model
for the data.6
Objective
function value
An objective
function value is the sum of squared deviations between the predictions
and the observations. Depending upon the estimation methods (Ordinary
least squares (OLS), Extended least squares (ELS), Weighted least
squares (WLS)) the objective function is appropriately weighted.
In NONMEM, The objective
function is not a sum of squares, it is -2 times the log of the
likelihood. The likelihood, in simple normal problems is a sum of
squares.
Optimization is a process
of finding that estimate of a parameter, that maximizes or minimizes
a given function.7
For e.g.: In a maximum likelihood approach, it is that value of
the parameter which maximizes the objective function value. Different
softwares use different optimization algorithms (Newton-Ralphson,
Simplex, Nelder-mead, E-M algorithm etc.,) Optimization invariably
proceeds by iteration. Starting from an approximate trial solution,
a useful algorithm will gradually refine the working estimate until
a predetermined level of precision has been reached. If the function
is smooth, an optimization algorithm can be expected to converge
to a maxima or minima when given a sufficiently good starting value.
A good starting value is one of the keys to success.
The diagram below shows
how an optimization algorithm (golden search method) converge to
either a local minima or global minima of a function.
When the objective function
value is computed as -2 times the log of the likelihood, competing
nested models (covariate model and no covariate model) are compared
by calculating the difference between their objective function value
referred to as log-likelihood ratio.
The log-likelihood ratio is assumed to be Chi-squared distributed.
A difference in objective function value of 3.84 is considered to
be significant at p<0.05 with one degrees of freedom (difference
of one parameter between the two nested models).
A single
numerical value used as an estimate of a population parameter. For
e.g.: population mean. A point estimate does not provide any information
about the inherent variability of the estimator.
Confidence
interval
An interval
that estimates a population parameter within a range of possible values
at a specified probability.The confidence interval reflect the uncertainty
associated with the parameter estimate or in other words, the precision
of a parameter estimate.
A 95% confidence
interval for a population mean can be described in the following manner 9.
When a clinical trial is repeated 100 times, and if we construct 100
confidence intervals out of those trials, 95 of those confidence intervals
would contain the population mean and 5 of the intervals would not.
Caution:
A 95% confidence interval of a mean is not to be defined as
95% of the population mean lie in the interval, nor 95% of the times
the population mean is contained in the interval.
- Most mixed effects
modeling softwares provide the asymptotic standard error's (S.E),
which can be used to derive confidence intervals.
- These S.E's are estimated
assuming the possible values are normally distributed, which may
not be applicable to variances or Emax function. i.e., Maximum
value for Emax = 1, a confidence interval greater than one does
not make sense.
Maximum
a posteriori / Posthoc estimates
The prior
distribution of the parameters across a population and the actual
data of the individual are used to obtain the posterior probability
of individual parameter estimates.These estimates are called maximum
a posteriori or posthoc estimates.
Simulation
Simulation is the use of
the model to predict future outcomes, such as drug effect after an
altered dosage regimen. Simulation can be braodly classified into
deterministic and stochastic simulation. 8
1. Deterministic simulation
These simulations are
performed using a 'simulation model' , with no stochastic (statistical)
sub models. They consider only typical values of the parameters
and not the entire distribution of the model parameters. e.g.: role
of dosage regimen in controlling pharmacodynamic responses.
2. Stochastic simulation
These simulations are
performed using a 'simulation model' with the stochastic sub-models.They
consider the interpatient, inter occasion and/or residual variability.e.g.:evaluation
of equivalence intervals of highly variable drugs.
Stochastic simulations
can be classified into
a.
Parametric (montecarlo)simulations
These simulations
are performed by resampling from one or more parametric distribution
models, for e.g.: sampling the clearance of the drug given the
mean and variance of a log normal distribution.
b. Non-parametric simulations
Simulations are performed
by resampling from sets of observed data, as opposed to using
a distribution fitted to these data, for e.g.: sampling the
clearance of a drug from a set of clearances using the bootstrap
techniques.
Clinical
trial designs
The most widely used clinical
trial designs to evaluate effectiveness and safety of a drug are parallel,
cross - over, and titration designs.
o Parallel
In a parallel design
trial, patients are randomized into cohorts who receive one of
the several treatments (control, dose 1, dose 2 or dose 3). ).
Such a design will offer the population, rather than the individual,
PK / PD characteristics. The advantage of such a design is the
lack of confounding factors such as time (carry over effects)
and design dependent outcomes. An illustration of parallel design
is given below: 
o Crossover
In a crossover design,
each patient receives some (incomplete block) or all (complete
block) of the treatments being studied.Therefore, a cross - over
design is the most powerful design if deducing the individual
concentration (or dose) - response curves is the ultimate aim.
In a crossover design, three types of effects are evaluated. They
are period effect (Period I vs Period II), Carry over effect and
sequence effect (Placebo, active treatment or active treatment
, placebo).The disadvantages of this approach are that of its
longer trial duration, possible carry - over effects from previous
doses and the need for sophisticated data analysis (nonlinear
mixed effects modeling). An illustration of crossover design is
provided below:
The
titration design ensures that the patients usually start at a
relatively low dose and the dose is increased gradually until
either no additional benefit is observed or dose - limiting toxicity
occurs. This design closely resembles the clinical practice and
the individual PK / PD characteristics can be obtained. The major
disadvantage of this design is that of the possibility of an inverted
U-shaped PK / PD relationship, as an artifact. The patients who
are less sensitive to the drug need higher doses of the drug,
making it appear as if the response decreases after a certain
dose. Data analysis using conventional methods such as ANOVA fail
and the use of sophisticated modeling techniques is required
Pharmacodynamic
terms
o Response
Response is any entity
of a system that can or is measured. e.g.: Glucose concentration,
forced expiratory volume.
o Effect
Effect is calculated
as difference between the baseline response and maximum/minimum
response. e.g.: % decrease in blood glucose concentration.
o Efficacy
The maximum effect
that can be produced by a drug.
o Potency
A potency of the drug,
is the dose at which 50% of maximum effect is attained.
o Effectiveness
Effectiveness is a
simple "yes" or "no" answer to show that the drug works or not.
References
1.
Link to statistics glossary, Accessed March 20,2005
2.
Definition for model, Accessed March 20,2005
3.
Sheiner & Steimer, 2000,Pharmacokinetic/pharmacodynamic modeling
in drug development. Annu Rev Pharmacol Toxicol. 2000;40:67-95
4. Grasela TH, Sheiner
LB., "Pharmacostatistical modeling for observational data," Journal
of PK & Biopharm 1991: 19(3) (June Supplement):11S-23S
5. Aarons, L., "Population
Pharmacokinetics: Theory and practice," Br J Clin Pharmacol
1991; 32: 669-670
6. Maximum
likelihood, Accessed March 20,2005
7. Optimization
procedure
8.Gobburu
et al, 2001,Utilisation of pharmacokinetic-pharmacodynamic modelling
and simulation in regulatory decision-making. Clin Pharmacokinet.
2001;40(12):883-92
9. SB Tan, Introduction to Bayesian Methods for Medical Research. Ann Acad Med Singapore.2001;30:444-446
back
to top
|
