Oct 19, 2005 - However, if one find only 2 points in elimination phase, it is always good to go for model-fitting. Software which allow adequate graphical presentations of the data as. For example, in WinNonlin the following algorithm is used. (intercept, slope and residual variance). If you have 2.
PharmPK Discussion List Archive - PK2005095.html - 2005 PharmPK Discussion - Calculation of half-life. On 19 Oct 2005 at 11:27:16, dsharp5.at.rdg.boehringer-ingelheim.com sent the message The following message was posted to: PharmPK Group: This is one of those very simple questions because it is very basic and therefore gets lost in sophisticated discussions that we have in this group.
In my CRO and consulting days I worked with a number of companies on PK. I found a number of them allowed the computation of elimination rate constants (and half-life and AUCinfinity) using only two points in the terminal phase. Of course their r-squared values are quite good(!), but I never believed this was a legitimate approach, although I failed in convincing them of this.
If one consults standard texts they say a minimum of three points is needed, but why? Searching my memory back in the hazy mists of the past, it strikes me that it requires 3 points to uniquely define an exponential function. When we do a log transform the resulting straight line requires only two points, but we shouldn't lose sight of the fact that it's an exponential function we are determining. My question is, is the use of at least three points a mathematical necessity, or merely good sense? If it is the latter, than good sense obviously differs from place to place.
I am not formally trained in PK (or anything else I do for that matter!) so I missed this early lesson. Please edify me. I have considered a post entitled 'Stupid PK tricks' where I outline some the dubious approaches I have 'experienced', but it would only be for humor, and would not fit with the serious nature of the group. Dale Standard text? Two points is the minimum for half-life with the assumption that you are taking two points from the log-linear terminal phase. Three (or more) points allows you to start testing/ verifying that assumption.db. On 19 Oct 2005 at 13:17:19, kevin.m.koch.at.gsk.com sent the message Dale, If we linearize the exponential function, it can be defined by two points.
It's just good sense to use more. But more important than the number of points is the time frame over which they are spread.
Two or three points spanning a couple of half-lives should better estimate the elimination function than a dozen points spanning a fraction of a half-life. Kevin. On 19 Oct 2005 at 14:59:27, Xiaodong Shen (shenxiaodong11.-at.yahoo.com) sent the message The following message was posted to: PharmPK Hi, To judge if it is a line we need at least three points, two points always make a line in terms of mathematics. In addition, I am not a PK person.
Xiaodong. On 19 Oct 2005 at 19:11:38, Indranil Bhattacharya (ibhattacharya.-a.gmail.com) sent the message Dale, from my limited experience in the world of PK, I would suggest that three points should be considered for estimation of elimination half life.
My justification for the selection being 1) that with more than two data points I will have a 'richer' data set to compute the elimination half life and the half life calculated would be a 'better estimate'. 2) At lower concentrations I would expect more variability (due to the assay) assuming that profile is being followed until it reaches LOQ. Of course the selection and overall contribution of the third point depends upon its position in the PK profile. Indranil Bhattacharya Ph.D candidate Dept. Of Pharmaceutical Sciences State University of New York at Buffalo Usa. On 19 Oct 2005 at 21:52:26, 'Kassem Abouchehade' (kassem.at.pharm.mun.ca) sent the message The following message was posted to: PharmPK Dale, when determining the half-life from the slope (-K/2.303) of terminal line resulting from a plot of log C vs time, we rely on at least 3 points which is more reliable. The time points should be selected such that the interval between the first and the last point chosen is more than twice the estimated half-life based on them.
Using two points will be less accurate and not reliable especially when dealing with drugs with very long half-lives and also depends how low the drug can be detected during the elimination phase. Also when comparing the terminal phases of two drugs one with long and the other with a short half-life, relying on two points only is not accurate and will not provide a fair PK comparison between the two drugs. Kassem. On 19 Oct 2005 at 23:44:37, 'Kassem Abouchehade' (kassem.at.pharm.mun.ca) sent the message The following message was posted to: PharmPK Dale, I would like to add also this old paper by Gibaldi and Weintraub for your reference: Gibaldi M, Weintraub H. 1971 Apr;60(4):624-6.
'Some considerations as to the determination and significance of biologic half-life'. Kassem. On 19 Oct 2005 at 21:30:53, Varma MVS (varmamvs.at.yahoo.com) sent the message HI, Reliable Kel needs use of more then 2 points from the terminal profile. Although a straight line can be drawn with 2 points, that makes no sense satistically. In many practicle situations the terminal portion of Plasma conc profile falls very close to LOQ where the analyticl variability is maximum. Thus considering last and lastbut one points will lead to worng numbers. Instead averaging the Kel obtained with subsequent points of atleast 3 points will give a better picture.
However, if one find only 2 points in elimination phase, it is always good to go for model-fitting or non-compartmetal analysis. Varma Manthena. On 20 Oct 2005 at 08:34:38, 'Willi Cawello' (Willi.Cawello.at.schwarzpharma.com) sent the message Dear Dale, I expect your question refers to the terminal half-life. Under this condition please find this anser: The working group pharmacokinetics of the AGAH (Association for Applied Human Pharmacology) has published the results of thier discussions about PK items in a text book (Parameters of Compartment- free Pharmacokinetics, Willi Cawello (Ed.), 1999). Please find an extract from section 4.2.1 titled 'Calculation of the terminal half- life from plasma data': In general, only the terminal half-life is determined by model- independent methods. Conceptually, this is carried out by means of a semilogarithmic presentation of measured drug concentrations versus time. In order to decide whether calculation of a half-life is meaningful, the terminal portion of this presentation has to be examined.
If the data in this portion of the profile can be reasonably well approximated by a straight line, a (terminal) half- life t1/2 can be calculated according to t1/2 = ln (2) / lambda-z F4.7 where lambda-z denotes the slope of the approximating straight line. Calculation of lambda-z is generally carried out by unweighted linear regression Snedecor and Cochran, 1989 resulting in lambda-z = sum(ti). sum(ln Ci) - n. sum(ti) ln Ci / n. sum(ti^2) - (sum(ti))^2 F4.8 where n is the number of data points used in the regression analysis, ti the respective times and ln Ci the corresponding logged drug concentrations (to base e). There are no fixed rules for the selection of data to be used in this analysis, but the following hints may give some guidance: 1. As far as possible, all concentration data in the terminal phase should be selected; however, a minimum of three data points should be used.
Whenever possible, the last concentration measured at the end of the profile should be used. Taking this concentration into account could be problematic for cases in which it is higher than concentration values at earlier time points (including values lower than the limit of quantification (LOQ)). The maximum observed drug concentration, Cmax, should only be used if it is not substantially affected by drug absorption. From a practical viewpoint, the determination of half-lives is best accomplished by means of interactive pharmacokinetic or statistical software which allow adequate graphical presentations of the data as well as corresponding calculations of pharmacokinetic parameters, such as the terminal half-life t1/2. As a general rule, the observation period should be about three to five times of the supposed half-life and five observations should be scheduled within the range of the terminal phase. For example, if the supposed half-life is 8 hours, blood samples should be collected up to 24 - 40 hours after drug administration, with samples taken e.g. At 10, 12, 16, 24 and 36 h.
B.) Using more sophisticated methods (so called peeling methods or methods of residuals) it is possible to determine not only the terminal half-life but also the half-lives described in equation C(t) =A1.exp(-lambda1.t)+A2.exp(-lambda2.t)+. Gibaldi and Perrier, 1982.
C.) The half-life of a drug can show large interindividual variability. D.) Each individual drug concentration vs. Time profile should be evaluated separately. For reasons of consistency, it is recommended to initially present all the profiles together on a semilogarithmic scale and to consider the following questions: Is it possible to use all the plasma concentrations following a timepoint common to all the profiles? Is it possible to use all the plasma concentrations within a given time window (e.g.
From 4-12 h after drug intake)? Is it possible to use the last n drug concentrations for each profile (n 0xB33)? E.) Alongside these graphical-based methods for determining half- lives, other methods based on mathematical algorithms are also available. For example, in WinNonlin the following algorithm is used: Linear regressions are repeated using the last three points, the last four points, the last five points etc.
For each regression, an adjusted R2 is computed: where n is the number of data points in the regression and R2 is the square of the correlation coefficient. The regression with the largest adjusted R2 is selected to estimate the terminal half-life, with one caveat: if the adjusted R2 does not improve, but is within.
0001 of the largest value, the regression with the larger number of points is used. Best regards, Willi. On 20 Oct 2005 at 17:24:03, Stephen Duffull (steveduffull.-at.yahoo.com.au) sent the message Hi all I think there are 2 distinct components to this discussion: 1) How many data points do you need to estimate the parameters of a straight line and 2) How many data points do you need to estimate the log-linear slope in a PK noncompartmental study. You need 3 points. There are really 3 parameters (intercept, slope and residual variance). If you have 2 parameters then you assume incorrectly that there is no residual variability.
I would think that there must be some guidance on this. Regards Steve Point 1. Interesting, so you want to know how good your parameter estimates are as well, or is this just an estimate of fit? Two points seem to be sufficient for our clinical colleagues, maybe they assume residual variance is the same (similar) from case to case and don't need to estimate it every time they draw blood samples. Reminds me of the time a well respected colleague presented data with a straight line drawn through one point, he had assumed the slope;-) - db. On 20 Oct 2005 at 09:38:00, 'J.H.Proost' (J.H.Proost.aaa.rug.nl) sent the message The following message was posted to: PharmPK Dear Dale, I agree with several comments pointing to the importance of the concentration range, in terms of half-lives, for the precision of the estimated elimination rate constant (k) and half-life. I'm not really happy with the suggestions that three data points can be used for the estimation of k.
It is good practice to calculate the standard error and confidence interval of the estimate of k. This gives a good (although certainly not perfect) idea of the reliability of the calculated value of k. With two points the standard error is infinite. Please note that one should use the t-distribution for the calculation of the confidence intervals, and not the normal distribution. With three data points the t-value for the 95% confidence interval is 12.7 (one degree of freedom), so the confidence interval is very wide.
With four data points the t-value is 4.3 (two degrees of freedom), and the confidence interval is much less wide. For more data points the gain in precision is not so spectacular (t = 3.2 for five points), so four data points seems a reasonable minimum value.
A second comment refers to the purpose of the estimation of k. If it is used for the estimation of the AUC from the last time point to infinity, and the extrapolated area is relatively small compared to the total AUC, the precision of k is not really a major topic, and a two-point estimate may be 'good enough'. In that case it is not the confidence interval of k that matters, but the confidence interval of the estimated total AUC. Best regards, Hans Proost Johannes H. Of Pharmacokinetics and Drug Delivery University Centre for Pharmacy Antonius Deusinglaan 1 9713 AV Groningen, The Netherlands tel. 31-50 363 3292 fax 31-50 363 3247 Email: j.h.proost.at.rug.nl.
On 20 Oct 2005 at 09:50:30, andreanicole.edginton.aaa.bayertechnology.com sent the message Dear group: An increase in the resolution of points along the 'terminal phase' will affect the calculation of half-life. The terminal phase can be weakly defined by the last two data points. As points are included between the last two time points (usually relatively far apart) the likelihood of detecting an additional 'terminal phase' increases. As long as the elimination is first order, taking the two point approach will likely underestimate half-life.
Increasing the number of points to three is indeed superior. Andrea - Bayer Technology Services GmbH Process Technology, Biophysics Leverkusen, Germany. On 20 Oct 2005 at 13:36:01, =?ISO-8859-1?Q?HelmutSch=FCtz?= (helmut.schuetz.aaa.bebac.at) sent the message The following message was posted to: PharmPK Hi Dale!
I found a number of them allowed the computation of elimination rate constants (and half-life and AUCinfinity) using only two points in the terminal phase. Of course their r-squared values are quite good(!). With only two points it must have been not only /good/, but.exactly. 1.
![How to find intercept from winnonlin software How to find intercept from winnonlin software](http://www.wikihow.com/images/0/02/Calculate-Slope-and-Intercepts-of-a-Line-Step-6.jpg)
Searching my memory back in the hazy mists of the past, it strikes me that it requires 3 points to uniquely define an exponential function. When we do a log transform the resulting straight line requires only two points, but we shouldn't lose sight of the fact that it's an exponential function we are determining. No, since 1 y = A. exp(B. x) contains.two.
parameters, two points also suffice for the exponential. The only difference is, that the transformed equation 2 ln(y) = ln(A) + B. x can be solved directly through a set of linear equations, whereas 1 is nonlinear in parameter B and therefore calls for an iterative procedure. You can check this with wonderful M$-Excel: A=100, B=-ln(2)/12=-0.4666210 (half-life = 12) x= 0 y=100 x=12 y= 50 applying a linear regression to x ln(y) (i.e. 1) gives A=1000000, B=-0.4666220 whereas the built-in 'Solver'-routine (i.e. 2) gives A=1642800, B=-0.3824150 Turning the screws (e.g., changing the number of iterations, the sensitivity, etc.), different values will be obtained. If you change the sign of parameter B in the models to y = A.
exp(-B. x) and ln(y) = ln(A) - B. x you will get A=1000000, B=0.4666220 (LR) A=1952850, B=0.2517360 (Solver) This simple example shows, why 1 rather than 2 is applied in 'non-compartmental' PK.
As David and Xiadong already pointed out we need at least three points to look for linearity (since with two points we have zero degrees of freedom for testing). There was a rather long thread about R2 in 2002, you may have a look at or if the link is not working, go to the search page with the key-words 'Non-compartmental' 'Analysis' 'Odeh' best regards, Helmut - Helmut Sch=FCtz BEBAC Consultancy Services for Bioequivalence and Bioavailability Studies Neubaugasse Vienna/Austria tel/fax +43 1 2311746 Bioequivalence/Bioavailability Forum at The archive page URLs change from time to time. When ever I redo an yearly archive the URLs may change. For the current year this can be quite often.
Sometimes I change my archive software and redo all the archives. The last time was when I added some extra munging of the email addresses in the archive (see mungfaq.html). Helmut's search terms work exactly but a more general approach is to use the title/topic as a search term. With title/topic and year you can look up the entry on the annual index at www.boomer.org/pkin/ - db. On 20 Oct 2005 at 08:02:56, Xiaodong Shen (shenxiaodong11.-at.yahoo.com) sent the message The following message was posted to: PharmPK Hi, Even two points always give r-squared value 1, 1 makes a line looks very good.
But people would never use two points to judge if it is a line since with two points you can only draw one line and also a very straight line. Xiaodong. On 20 Oct 2005 at 13:32:19, dsharp5.aaa.rdg.boehringer-ingelheim.com sent the message The following message was posted to: PharmPK All, Thank you very much for your comments. My own personal practice is very similar to what Willi outlined, however, I have not always been successful convincing others that this is the best approach.
If, as Johannes has suggested, which is the SE of Kel of a 2-point line is infinity, than I would say this not useable. A two point terminal phase tells us that the true kel value is somewhere between + and minus infinity. I would maintain we knew that without running any experiments.
I believe this may be another way of stating my argument, which that infinitely many exponentials can be drawn between 2 points. Certainly no one would argue against the idea that more points in the terminal phase are better than fewer points, but oftentimes in animal studies blood volume and animal care considerations mandate the collection of fewer samples. My approach for profiles with only two points in the terminal phase is report AUClast, Cmax and Tmax and not go any further. Nonetheless, what is the consensus of the group?
Is the use of two point terminal phases mathematically proscribed, or merely good sense. Should we accept the results of this analysis? I can point to a literature paper or two where TK based on 2 points in the terminal phase was reported, so it gets by some referees. On 20 Oct 2005 at 20:06:08, =?ISO-8859-1?Q?J=FCrgenBulitta?= (bulitta.at.ibmp.osn.de) sent the message The following message was posted to: PharmPK Dear All, In addition to the points already mentioned, it might be worth adding a 'bioequivalence point of view', especially for extended release formulations. I think the number of points used to derive the terminal half-life really should be chosen based on a specified objective for the drug under discussion. As Dr Proost pointed out, what really matters is the impact of the uncertainty in estimated terminal half-life on the parameter of interest.
Among others, AUC0-infinity, AUMC (!), MRT, Vss, Vz, and T1/2 itself. If one is really interested in the influence of the choice of the number of data-points on the bias and precision in terminal half-life and its derived parameters, a simulation approach for different proportional and additive analytical errors with subsequent non- compartmental evaluation might be a reasonable choice.
This approach might be considered to determine, if the chance to show bioequivalence is affected by the method of estimating terminal half- life, e.g. For a drug with a long half-life and a difficult analytical assay. My personal practice: I usually use 3-6 datapoints (for some drugs 4-6) to estimate terminal half-life based on visual inspection (e.g. In WinNonlin) and R^2-adjusted. If the assay precision is good and if there is a systematic increase (or decrease) the more points are selected, I choose 3-4 points. Only if the third point is Cmax, then I go for 2 points or skip estimation of T1/2 for this subject. Hope this helps.
Best regards Juergen - Juergen Bulitta Scientific Employee, IBMP Paul-Ehrlich-Str. 19 D-90562 Nuernberg Germany. On 21 Oct 2005 at 00:11:29, (Kees.Bol.aaa.kinesis-pharma.com) sent the message The following message was posted to: PharmPK Dear, After reading a couple of messages I think the approaches are sometimes too scientific, and not practical enough. In standard pharmaceutical PK reports one does not report SE and CI on the estimation of k. One estimates k, mostly on the basis of a minimum of 3 data-points.
Acceptance of the estimates is based on other criteria, e.g. R^2 is at least 0.9 (differs from company to company), and the time-span of the data-points used in the calculation should be at least 2x the estimate of T1/2 (one of our criteria). At the end it doesn't really matter if the estimation of your T1/2 is 12, 10.5 or 13, because for one subject you wil overestimate t1/2 for the other you will underestimate T1/2. What will be the focus of many reports is the mean or median T1/2 and the intersubject variability. If your sample size is large enough, your mean or median estimate will not differ much if you use different criteria (as long as your criteria are predefined and consequently used). If the purpose of your trial is to formally compare two treatments statistically a poor estimation will increase your intersubject variability, and may require a larger sample size. What you could also do is improve the design of your study, e.g measure longer, improve the sensitivity of your bioassay.
In toxicokinetic studies you often have the problem that you can not measure the concentrations long enough because you hit the LOQ much quicker (metabolism is often much faster in rats, mice etc.), or that you are not able to take enough blood samples without bleeding the animal too much. As a result you sometimes have studies in which you only have 2 data-points in the terminal phase in almost every animal. Then again you have to be practical (because you don't want to, or don't have the resources, to do population PK for every preclinical study). You still calculate T1/2 and report the mean or median, but give a remark that T1/2 and the related paramters could not be estimated accurately.
At least you have learned something from your study. You know that the T1/2 was say about 10 hours and not 2 hours or 100 hours. The above methods have been used in many drug filings to regulatory authorities.
They may not be that scientifically sound to some of you, but at least it helps you to move forwards. Best regards, Kees Kees Bol Kinesis Pharma BV Consultants in Drug Development The Netherlands. On 21 Oct 2005 at 13:58:45, 'Hans Proost' (j.h.proost.-a.rug.nl) sent the message The following message was posted to: PharmPK Dear all, Kees Bol wrote: In standard pharmaceutical PK reports one does not report SE and CI on the estimation of k. OK, but why should one not improve the 'standard' PK report? And it is not really required to report SE and CI; these values can be used to judge whether or not the estimation of k is sufficiently precise to report. If not, this should be reported.
This refers to any value mentioned in a report. Acceptance of the estimates is based on other criteria, e.g.
R^2 is at least 0.9 (differs from company to company), What is the rationale of this criterion? As I have written in earlier message, R^2 (or 'adjusted R^2') is not a suitable criterion for goodness-of-fit. Among others, because it does not take into account the number of data points used (remember that R^2 is exactly 1 for two points). Willi Cawello wrote: e.) Alongside these graphical-based methods for determining half- lives, other methods based on mathematical algorithms are also available. For example, in WinNonlin the following algorithm is used: Linear regressions are repeated using the last three points, the last four points, the last five points etc. For each regression, an adjusted R2 is computed: where n is the number of data points in the regression and R2 is the square of the correlation coefficient.
The regression with the largest adjusted R2 is selected to estimate the terminal half-lifewith one caveat: if the adjusted R2 does not improve, but is within. 0001 of the largest value, the regression with the larger number of points is used. Is there any scientific proof of this approach? Taking into account the aforementioned property of R^2 I doubt whether this is a valid approach. I would suggest a different approach, although I must admit that I did not proof this approach: Use the residual variance as the criterion for choosing the number of data points. The residual variance is the sum of the squared deviations (in the logarithmically transformed scale) divided by the degrees of freedom, i.e.
Please note that this is a suggestion only. I don't say that this approach is scienfically proven, and I don't say it is optimal. But at least it takes into account the number of data points in a plausible manner. Any comments are welcome! Best regards, Hans Proost Johannes H.
Of Pharmacokinetics and Drug Delivery University Centre for Pharmacy Antonius Deusinglaan 1 9713 AV Groningen, The Netherlands tel. 31-50 363 3292 fax 31-50 363 3247 Email: j.h.proost.-at.rug.nl. On 22 Oct 2005 at 14:56:41, 'Sima Sadray' (sadrai.-at.sina.tums.ac.ir) sent the message The following message was posted to: PharmPK Dear All, Here you can follow the discussion with true data. For Diclofenac we saw multiple peak phenomena so for some subjects we had only two point for k estimation. So we compare the two method (slope with two or three point) as you see the results may be very different.
There was underestimation for k and overestimation for t1/2 in this case.
Hi John, The Exponential Term Estimates A1 and Alpha1 are the intercept and slope that will be used to perform the numerical deconvolution and they equate to Czero and K10 in a 1 com model after IV bolus dosing. So you have to do one more step, simulate your data with the same Dose, V and K10 as before but with an IV model. Then you can read Czero form the Predicted data tab, don't forget that a) this is assuming 100% Bioavailability and you should dose normalise the intercept to be sure your Fabs in deconvolution is correct Simon.
Hi Simon, Thank you. One question, what do you mean by 'dose normalize the intercept to be sure your Fabs in deconvoluton is correct'? John sdavis wrote: Hi John, The Exponential Term Estimates A1 and Alpha1 are the intercept and slope that will be used to perform the numerical deconvolution and they equate to Czero and K10 in a 1 com model after IV bolus dosing. So you have to do one more step, simulate your data with the same Dose, V and K10 as before but with an IV model.
Then you can read Czero form the Predicted data tab, don't forget that a) this is assuming 100% Bioavailability and you should dose normalise the intercept to be sure your Fabs in deconvolution is correct Simon. John, imagine you had some Oral tablet of 20mg, if normalise this to 1mg then you would expect Czero to be one-twentieth - this is the UIR or Unit Impulse Response you may see in some of the manuals. If you use this as the 'A' then when you enter other dose amounts e.g. 15mg for other formulations then the scaling to get the correct Fabs is done for you, rather than you having to calculate the ratio each time. Is that clearer?
(PS it's only necessary to quote the part of the text you want me to comment on, not the whole message - then we can keep the threads easier to review and index).