Back to the Top
Dear All,
I have a very basic question that confused for long time. Hope someone can help me better understand
standard deviations of PK parameters in PK modeling.
If you fit mean drug concentration versus time data (only one curve) to one-compartment open model
(oral administration) using nonlinear regression method, you will get the best estimates for PK
parameters including, V, ka and kel. Also from the software, the standard deviation for these
estimates PK parameters will be provided. What do these standard deviation really mean? How are they
calculated? Are they statistically meaningful?
On the other hand, if you use individual curves (drug concentration vs time) do the same thing, you
also will have the standard deviations for these PK parameters. How are they calculated in this
case?
In these two cases, the standard deviations for pk parameters are both calculated. How to interpret
these values? What's the difference between them?
Many thanks
Maria
Back to the Top
Maria,
The standard deviations that you asked about are indeed different. In
the first case (fitting the mean data) standard deviation refers to the
confidence that you have about parameters that describe the mean curve.
You may construct confidence intervals based on these SD and from those
estimate precision the parameter estimates for the mean curve. This
should be OK for preclinical studies where all subjects are very similar
(and sampling is often destructive).
In the second case, you study many individuals. Presumably, you have
many data points for each individual so that you can estimate each
parameter for each individual. When you study distribution of these
parameters across study subjects, you are likely to see some
variability. Standard deviation of the parameter distribution will tell
you about variability within the study population rather than
characterize the precision of the mean curve. In some cases you may use
these parameter distributions and estimate mean parameters (as means of
individual parameters) and precision of the mean (as SD/sqrt(n-1)). This
precision is somewhat similar to the first case and characterizes the
confidence in the means of the parameter distributions. If the
distributions are log-normal, you take the log of the parameters,
perform the same computations and then exponentiate (thus computing
geometric mean and CI of the geometric means).
Yet another, more mathematically precise procedure is to use population
(nonlinear mixed effect) modeling. By fitting population model to the
data you get both the estimates of inter-subject variability (random
effects) and the typical ("mean") parameters together with the precision
of the parameter estimates. You may google for "population modeling" or
something similar to get tons of references, for example
http://www.nature.com/psp/journal/v1/n9/full/psp20124a.html
Leonid
Leonid Gibiansky, Ph.D.
President, QuantPharm LLC
web: www.quantpharm.com
Back to the Top
Maria,
There are two things you should try to understand about PK parameters - variability and uncertainty.
1. A PK parameter such as clearance will vary from person to person. If you have a collection of
clearance values each one from a different person then you can describe the variability by the
standard deviation of the collection of values. This is easy to do and explained in any standard
textbook of statistics. The variability of a parameter is useful for describing drugs which have a
lot of variability e.g. morphine compared to drugs with much less variability e.g. busulfan.
2. When a PK parameter is estimated there is always some uncertainty about the true value because of
things like measurement error and timing of samples. Each PK parameter estimate may be associated
with a measure of uncertainty called the standard error. Note that the term for this is not standard
deviation. The calculation of the standard error can be done in a variety of ways and it probably
doesn't matter which method is used because the standard error is only a crude guide to the
uncertainty. A better guide is a confidence interval calculated using a bootstrap method. The
uncertainty is probably most useful as a guide to how badly designed the PK study was. Big
uncertainty in a parameter means a bad design. Unfortunately this is quite common because people
don't think carefully when planning studies.
If you software reports individual parameter estimates with a standard deviation instead of a
standard error I would recommend not using it. Get software that understands that clearance is a
primary parameter and kel is a secondary parameter and knows the difference between variability and
uncertainty.
Best wishes,
Nick
--
Nick Holford, Professor Clinical Pharmacology
Dept Pharmacology & Clinical Pharmacology, Bldg 503 Room 302A
University of Auckland,85 Park Rd,Private Bag 92019,Auckland,New Zealand
email: n.holford.at.auckland.ac.nz
http://holford.fmhs.auckland.ac.nz/
Back to the Top
Hello Maria,
When we use any of the standard software ,Statistics calculated and displayed in the output while
fitting the PCT to any of the model and estimating the model parameters are Standard Error (SE) and
%CV of each parameters.
Why do we need these stats?
Accuracy and Precision of parameter estimates are understood from these statistical estimates.
Accuracy deals with p-hat in relation to true parameter value and precision deals with %CV
Estimated model parameters p-hat have no value unless they have a fair degree of precision. This can
be assessed by computing each parameters coefficient of variation
%CV=(SE/p-hat) *100
A large parameter CV doesn't imply that model is incorrect but it may be due to not enough samples
or not having samples at the appropriate times. For ex when a model contains several parameters
V=1000± 30L, CV=3% & K= 0.01±0.005 /hr, CV=50% then CV is a better way of expressing the
variability, because CV is a relative measure, where as SE is an absolute measure
After the model is fit when the C hat (predicted conc's) are displayed SD of each of these predicted
Conc at each of the time points are calculated,along with SD residuals also are important to
qualitatively asses the goodness of fit to the selected model, additionally SD is used to calculate
the Variance covariance matrix
in the model fitting
Regards
Kaushik
Back to the Top
Hi Maria,
I try to share my thoughts on this problem:
In the first scenario, you got parameter value, RSE%(CV) and 95% CI when
you run individual parameter estimation. They are calculated based on the
residual error (Y-Yhat) and independent variable(X) and related to the
non-linear function. They represent how well the model fits the data and
where the coefficients mostly likely are. Imagine the simple case of linear
regression to see how the SEs of alpha and beta are calculated.
In the second POPPK scenario, total variability is separated into RUVs and
BSVs. RUVs are unexplained variability. BSVs are explained by random
variability and structured variability(e.g. co-variates) in parameters. In
the output of ADAPT5, you got parameter value(pop mean), %RSE of parameter;
SD and SD's %RSE. SD represents the between-subject random variability as
you defined by ETAs. %RSE shows how well your parameters and SDs are
estimated.
The confusing outputs may have led to your question. The SDs in ADAPT5 is
actually refers to ETAs which represents random BSV. %RSE everywhere represents how well parameters
are estimated and relates to model fitting. It is related to RUV but not BSV.
Hope this helps!
Regards,
Jie
Back to the Top
Dear All,
Thank you so much for your answers. They are very helpful. I want to confirm whether I understand
are correct.
I am talking preclinical data in rats, and I don't have enough data to do a pop PK modeling.
For the first case, curve-fitting only performed for the mean data. There is no way to get the
variability information. The standard errors of pk parameters only indicate the uncertainty, and the
resulting intervals (for 95% CI) would bracket the true values of parameter in approximately 95% of
the cases, correct? If the values standard error is large, that means the model could not explain
the data very well, correct? It also could be due to not enough samples or not having samples at the
appropriate times, but it doesn't necessary mean the model is wrong.
Dr. Holford, you said
"The calculation of the standard error can be done in a variety of ways and it probably doesn't
matter which method is used because the standard error is only a crude guide to the uncertainty. A
better guide is a confidence interval calculated using a bootstrap method."
For only a set of data (only mean data), I still could not understand how the standard errors are
calculated. What do you mean the standard error is "a crude guide to the uncertainty". In this
case, there is no way to do bootstrap anyway.
I also confused about the calculation of SE for each parameter in the first case. It seems to me
very complicated. If you have different initial values for the parameters, will the SE of parameters
provided with software also change ?
For the second case, if it is a two-stage method, the standard error of each parameter should be
representative for the variability of these parameters, not the uncertainty. How to determine the
uncertainty of these parameters in this case, then?
Many thanks.
Min
Back to the Top
Dear Min,
The PKPlus module in GastroPlus provides the CVs for each fitted
parameter. It takes only a few seconds to fit 1, 2, and 3-compartment
models to a set of data and to be able to compare models both graphically
and to see their model parameter CVs, as well as the Akaike Information
Criterion and Schwatrz Criterion for each model. A variety of weighting
schemes are available as well.
Plots show absolute, log, and residuals with a mouse click for any model.
You can see more about PKPlus here:
http://www.simulations-plus.com/Products.aspx?pID=11&mID=11
If you want to send your CP-time data, I would be happy to provide the
results to you.
Best regards,
Walt
Walt Woltosz
Chairman and CEO
Simulations Plus, Inc. (NASDAQ: SLP)
42505 10th Street West
Lancaster, CA 93534
Back to the Top
Dear Nick et al:
It is interesting that all these discussions revolve around parametric estimates of model
parameter values. And the main points under discussion usually appear to be those of the mean and
standard deviation. I really wonder why this is. Why would anyone wish to make some assumption about
the shape of a model parameter distribution when it is not necessary? Why postulate a Gaussian or
lognormal or some other multimodal distribution when you can estimate the entire distribution right
away, using a nonparametric (NP) with no assumptions at all about its shape? Over the years we have
directly compared a parametric iterative Bayesian approach with the NP one. This is possible with
our IT2B software. The NP approach is not constrained by some parametric assumption. You might look
at
Bustad A, Terziivanov D, Leary R, Port R, Schumitzky A, and Jelliffe R: Parametric and Nonparametric
Population Methods: Their Comparative Performance in Analysing a Clinical Data Set and Two Monte
Carlo Simulation Studies. Clin. Pharmacokinet., 45: 365-383, 2006.
As I understand it, one also is interested in the various errors present in the data that is
analyzed. Many people assume an overall error pattern, having additive or proportional components,
and that is fine. But why not separate that error into its lab and clinical components? You can
easily do this. For example, our lab likes to first, determine the assay error over its full working
range before starting the pop modeling. Then, in addition, we like to estimate another parameter
which we call lambda, which is an additive noise term reflecting the errors with which doses are
prepared and given, the time errors associated with giving doses and drawing serum concentrations,
model misspecification, and possible changes in model parameter values (or distributions) during the
period of data analysis. What this does is to determine, first, the measurement error of the assay.
Then, second, it gives us an estimate of the noise associated with the clinical part of the study.
Small lambda, good precise clinical study. Big lambda, noisy clinical part of the study. This is
useful information, we think, and that is why we do it. Actually, these other terms are not
measurement noise at all, but are process noise which more properly should go in the differential
equations themselves.
About uncertainty in the estimates of the model parameter distributions - yes, bootstrap is the
best current method.
About software - just how does software "understand" that clearance is a primary parameter
while Kel is a secondary one? It has been obvious to many that they are quite interchangeable, and
one can use either one as desired. I am not as interested in a hypothetical volume completely
cleared of a drug as I am in the amount of drug transferred from one compartment to another in a
certain time. The mathematicians I know see no difference between the two, and I am told that there
are many texts on model identifiability that show this. You may like tomahtoes, I like tomatoes.
Chacun a son gout.
Respectfully,
Roger
Back to the Top
Dear Min,
Actually, it is possible to estimate variability from mean concentrations derived from a destructive
blood sampling study without using a mixed-effect model.
See Nedelman et al 1995, Pharm Res 12:124 which applies Bailer's method for obtaining confidence
intervals (and SEs) for AUC (when only one sample per subject but multiple subjects samples at
several sampling times).
Charlie
KinetAssist Ltd
Back to the Top
Hi Min,
Pop-pk method can be used for animal data even if there are not enough data points. In the extreme
case of one sample per individual, fixing the residual error is required to perform the analysis.
You would be able to get parameter, SD of parameter (intraindividual variability) and uncertainty
estimation for both. All are reported in results section.
Or you can use naive pooled data-taking the means of each time points and treat the data as "one
individual". You can get estimates for parameters and Uncertainty estimates of each parameter. Your
statements are right. And you are right that the parameters and uncertainty will change a little bit
with different initial values but not too much if your initial values are reasonable. The estimation
of function parameters are implemented through several different algorithms,e.g. Weighted least
squares, maximum likelihood etc. I believe there are mathematical equations in references to show
you how the uncertainty is calculated. Just remember you have data from one "individual " so there
is no intraindividual variation.
One thing to mention is that for AUC, CL etc. non-compartmental parameters and their
SD(intraindividual) variability can be estimated even for sparse sampling data. You can check the
help guide in Phoenix WinNonlin (sparse sampling). Briefly, AUC is calculated through the mean
concentrations and SD is calculated based on number of samples per time group, number of samples per
individual and variation within each time group. Or bootstrapping method according to the reference:
J pharm sci, 87, 372-387. (1998).
Hope this helps!
Regards
Jie
Back to the Top
Dear Jie, Min and others,
It is good to have this discussion. Here is my one-cent contribution. You propose to perform a naive
pooling approach on mean data. Why mean data? The correct way to do naive pooling is using all
individual data. Using mean data implies that you do a pre-analysis, and leave out relevant
information from the individual values, e.g. with respect to the variability. I'm quite sure that
this approach will provide more reliable standard errors and confidence intervals, e.g. by
bootstrapping. Moreover, I don't see any argument against using all individual data (in 2013 we
don't talk about longer runtimes, do we?).
best regards,
Hans Proost
Johannes H. Proost
Dept. of Pharmacokinetics, Toxicology and Targeting
University Centre for Pharmacy
Antonius Deusinglaan 1
9713 AV Groningen, The Netherlands
Email: j.h.proost.-a-.rug.nl
Back to the Top
Dear Roger et al:
The choice to use a normal distribution is not arbitrary. The normal distribution has the highest
entropy for all distributions with the same mean and variance. So even when the normal distribution
is the "wrong" distribution, it is still the best choice by virtue of being the "least worst".
Except in the case when you know apriori what the true distribution really is, but this doesn't
happen very often, especially for biological systems.
Sometimes distributional assumptions are necessary in the sense that some methods which we take for
granted nowadays cannot be done with the computing power available to mere mortals. As I understand
it current non-parametric methods require more computing power than parametric methods.
This brings me to something I don't understand about the non-parametric approach. I often see stated
that non-parametric methods make "no assumptions at all" about the shape of distributions. Does this
include "smoothness" of a distribution? It is a very innocuous and I think very reasonable
assumption that the true distribution of some complex natural phenomenon is smooth. But I am not
sure if this is taken into account in current non-parametric methods. Can you clarify?
Warm regards,
Douglas Eleveld
Back to the Top
Douglas,
The maximum entropy argument for assuming a normal distribution is not very compelling in the PK
world. The normal distribution is only the maximum entropy distribution for
random variables which have a range on the entire real line (both negative and positive values).
This is almost never the case for PK parameters, which are often known to be non-negative. Here the
the maximum entropy distribution is exponential (for a given known mean). To enforce
non-negativity, we usual alter the normality assumption to log normality, which is convenient
but not maximum entropy.
Two stronger arguments for use of a normality (or log normality) assumption are
a) it naturally leads to an extended least squares formulation for the corresponding likelihood,
which has certain 'good' properties and is often a reasonable objective function
even when the underlying random variable is not normal or log normal
b) it greatly simplifies the implementation of parametric EM methods
Bob Leary
Back to the Top
Hi Hans,
Thanks for your kind advice!
We had a special data set where there is only one sample per individual (destructive sampling of
animals). Assuming the individuals perform similar, we took the means at each time point
(naïve-pooled method) to generate a structure model for PK/PD. After establishing the structure
PK/PD model, pop-PK/PD method with all individual data then can be used to acquire more information
under the same model. I agree that Naïve-pooled method is limited by throwing away the information
of within time group variation. But it can be the initial step for model establishment and data
processing.
And I did try using all individual data and bootstrapping for estimating parameters such as AUC, CL
etc. non-compartmental parameters. However, further application of this method towards more
complicated functions such as CMT analysis or PK/PD analysis seems to be not guaranteed. Even the
authors of “destructive sampling bootstrapping method” had some concerns about estimating parameters
which are not linear combination of concentrations since “pseudo-individual” time profile is
generated and bootstrapped in the method. I would love to try the method but just wondering if
people believe it or not.
Thanks!
Regards,
Jie
Back to the Top
Dear Jie,
Thank you for your reply. A few comments from my side:
1) "we took the means at each time point (naïve-pooled method)"
In my view, the essence of the naïve-pooled method is that all data are pooled without considering
that they are obtained from different individuals (the aspect of 'destructive sampling' is not
relevant, as pointed out clearly by Nick Holford). Using the means instead of all individual data is
even one step more 'naïve'.
2) The use of means instead of all individual data may be a good starting point, but I don't see any
reason to use this 'naïve-naïve' method in a final analysis.
3) You say: 'further application of this method towards more complicated functions such as CMT
analysis or PK/PD analysis seems to be not guaranteed'. Is this your experience, or from the authors
of “destructive sampling bootstrapping method”? Do you have a reference?
best regards,
Hans Proost
Johannes H. Proost
Dept. of Pharmacokinetics, Toxicology and Targeting
University Centre for Pharmacy
Antonius Deusinglaan 1
9713 AV Groningen, The Netherlands
Back to the Top
Hi Hans,
Thanks for your comments!
I find out how to perform naïve-pooling using all data points, assuming them to come from one
individual. I had a wrong impression before. Thanks for catching that up.
For the 3) comment, I did consult with the authors and their reply was (if I may quote his email)
that"Using the (nested) bootstrap based on pseudo profiles we were able to estimate almost any
parameter derivable from non-compartmental kinetics and, most importantly, its variability. We did
not explore compartmental models, but I am sure it will become extremely complicated and will
probably deliver highly variable solutions, if any. In many so-called "rich data" situations already
2-CMT (input - central - peripheral) micro-parameters are difficult to estimate with sufficient
precision and
reproducibility (being at least robust in sensitivity analyses), in destructive sampling we would
not have any profiles. Under some reasonable assumptions I could imagine applications in PKPD. Using
simple pooled data will not provide closed-form variability estimates apart from linear functions of
C and I think the same holds true for PopPK (= nonlinear mixed models)"- from Dr. Harry Mager
Thanks!
Regards,
Jie
|
Want to post a follow-up message on this topic?
If this link does not work with your browser send a follow-up message to PharmPK@lists.ucdenver.edu with "Standard deviations for pk parameter estimates" as the subject |
Copyright 1995-2014 David W. A. Bourne (david@boomer.org)