Astea
★★

Russia,
2020-06-12 14:22
(797 d 08:22 ago)

Posting: # 21533
Views: 5,805

## Fantastic PK parameters and where to find them [NCA / SHAM]

Dear all!
While reading the forum I sometimes face to the statements that other PK parameters should be used in future researches for SD studies. Below is my collection of stange or rare parameters. I would be grateful if you'll comment on their properties, details of calculation and the perspectives of its using.

Cmax/AUC is referred sometimes to be more appropriate in BE studies than Cmax cause it has much lower variability (here we discuss Cmax/AUClast but not Cmax/AUCinf which is worse1). For decades it was used to be the third primary PK parameter in russian studies but several years ago it became unfashionable. In the absolute majority of studies it had no impact on the results until Cmax and AUC rules the decision. Even when bioequivalence was not proven for Cmax and AUC Cmax/AUC could be within the BE limits. The distribution type of the function is also questionable (ratio of log-normal - paranormal ?).

pAUC (partial AUC, truncated AUC), index of early exposure, as was discussed in the parallel thread could be more sensitive than AUClast ('pAUC was always more sensitive than Cmax'2). The question is what time should be the end time of the calculation? It could be a fixed value (like 72 h), Tmax or common3 Tlast, Tlow (AUClow, AUCbetween)... The time should be related to clinically relevant pharmacodynamic measure. What are their advantages?

AUMC (first order momentum, the area under the curve 'Concentration*time-time'). If we'll look at the physics analogy - the first-order momentum for the plain figure is the static momentum which defines the center of mass coordinates, while the second-order momentum is moment of inertia. The definition could be generalised to the n-th
order momentum (why to limit ourselves by first order? There could be second or third order as well...).

MRT is a most common PK parameters that compels to look at the AUMC. There existed approaches4 to use MRT instead of T1/2 in order to choose the appropriate washout period.

Capical was invented in order to explore the shape of the peak more precisely. As far as I understood it was first mentioned in an article in 19885 and was defined as the arithmetic mean of the concentrations within a 95% confidence interval of Cmax (i.e., 'not distinguishable from Cmax at the 5% level of statistical probability by the assay used'). The appropriate time tap lower and tap high, as well as apical duration were also discussed. The mean 20% apical time (arithmetic mean of the times associated with the concentrations within 20% of Cmax) and the appropriate time parameters were also mentioned in the article. In the later publications Capical was defined as the arithmetic mean of concentrations at some level (e.g.25% or 50% but not 95% CI!) below Cmax. I've heard that this parameter was figured out as one of the perspective PK parameters from SD studies while discussing the necessity of multiple dose studies in BE testing, but I could find only few articles that deal with it. If Capical is so promising parameter, why can't I find any trials with it in results (excepting [6] and a paper on controlled release gastroretentive dosage forms tested on dogs). Or my search methods are too poor? I would be grateful if anyone will give an example of it's calculation.

The concentration at the end of the intended dosing interval (Cτ) should serve as an analogue of Cmin with lack of multiple dose studies.

The lag time matters when we deal with delayed-release formulations. Note that Phoenix defines it as time of observation prior to the first observation with a measurable (non-zero) concentration but not the time to the first observation with a measurable concentration. There could also be other ways7 to define it.

T50%lower, T50%upper, T50%between (half-value duration) or T75%lower, T75%upper - parameters named as "Therapeutic response" ('Where it exists, consideration must be given to the "therapeutic window."')8. But aren't there limitations that could not afford to make a statement of any correlation between plasma concentrations and effect? The listed parameters can also be seen in the biosimilar studies.
Note that currently in Phoenix there are no direct possibility to calculate T50%late or T50%early as inter­sections.

MDT - midpoint duration time (midpoint of half-value duration). In the original article9 of Laszlo Endrenyi and Laszlo Tothfalusi the picture is listed in order to help to understand the meaning of the calculattion. The graph on the picture is pretty simple.
But may I have a question - do your really ever deal with such 'handsome curves'? Me - not.. My personal experience gives me an idea that the majority of individual curves look like "uncombed hedgehogs". How to define MDT for such a
curve for example (modified release or/and endogenous drug)?

I guess there may be no less than three opinions:
1). The calculation of MDT and it's interpretation for the following case is impossible
2). The calculation of MDT could be as the follows: t50%lower+1/2*t50%between, where t
50%between - time between the first and the last point where C=50% Cmax
3). The calculation of MDT could be as follows: t50%lower+1/2*t'50%between, where t'50%between - overall time below C=50% Cmax.

Usually listed strange PK objects pop up when we deal with modified release products. But what do you think about 'nice to knowing' them for IR products?

[1] Tothfalusi L, Endrenyi L. Without extrapolation, Cmax/AUC is an effective metric in investigations of bioequivalence. Pharm Res. 1995;12(6):937-42. doi:10.1023/A:1016237826520
[2] Vincze I, Endrenyi L, Tothfalusi L. Bioequivalence metrics for absorption rates: linearity, specificity, sensitivity. Acta Pharm Hung. 2019;89(1):17–21. doi:10.33892/aph.2019.89.17-21
[3] Fisher D, Kramer W, Burmeister Getz E. Evaluation of a Scenario in Which Estimates of Bioequivalence Are Biased and a Proposed Solution: tlast (Common). J Clin Pharm. 2016; 56(7): 794–800. doi:10.1002/jcph.663
[4] Grabowski T, Gad SC, Jaroszewski JJ, Guelen P, Deterministic chaos and wash-out determination in crossover trials, Int J of Pharmacokinetics, V. 1, N. 1, doi:10.4155/ipk.16.1
[5] Pollack PT, Freeman DJ, Carruthers SG. Mean apical concentration and duration in the comparative bioavailability of slowly absorbed and eliminated drug preparations. J Pharm Sci. 1988;77:477–80.
[6] Bialer M, Arcavi L, Susann S, Volosov A, Yacobi A, Moros D, et al. Existing and new criteria for bioequivalence evaluation of new controlled release (CR) products of carbamazepine. Epilepsy Res. 1998;32:371–8.
[7] Czismadia F, L Endrényi. Model-independent estimation of lag-times with first-order absorption and disposition. J Pharm Sci 87, 608–12 (1998)
[8] Skelly JP, Barr WH. Biopharmaceutic considerations in designing and evaluating novel drug delivery systems. Clin Res Pract Drug Reg Aff. 1985;3(4):501–39.doi:10.3109/10601338509051086
[9] Endrenyi L, Tothfalusi L, Metrics for the Evaluation of Bioequivalence of Modified-Release Formulations, The AAPS Journal, Vol. 14, No. 4, December 2012, doi:10.1208/s12248-012-9396-8

"Being in minority, even a minority of one, did not make you mad"
Helmut
★★★

Vienna, Austria,
2020-06-12 16:23
(797 d 06:22 ago)

@ Astea
Posting: # 21534
Views: 4,770

## Fantastic post 👍🏽

Hi Nastia,

limited time (bloody bread-and-butter job). Some desultory thoughts about PK metrics (more to come).

» Cmax/AUC is referred sometimes to be more appropriate in BE studies than Cmax cause it has much lower variability…

Not only that (it’s a side-effect). What are we interested in? Extent of absoption (cleary AUC) and rate of absorption (ka and possibly tlag). ka is not easily accessible in NCA. Cmax is a composite surrogate (because influenced by AUC). Easy to show: Define any PK model and vary ƒ whilst keeping ka and tlag constant. Cmax will change… Cmax/AUC is an attempt to deal with that.

» The distribution type of the function is also questionable (ratio of log-normal - paranormal ?).

Do you know 'Pataphysics? Seriously, László (The Younger) asked me the same question years ago, which I could not answer. Martin helped us out. It doesn’t matter: The sum/difference of two normal distributions will be normal, the same here: It will be log-normal.

» pAUC (partial AUC, truncated AUC), index of early exposure, as was discussed in the parallel thread could be more sensitive than AUClast ('pAUC was always more sensitive than Cmax'2). The question is what time should be the end time of the calculation? […] The time should be related to clinically relevant pharmacodynamic measure. What are their advantages?

The jury is out. E.g., for biphasic methylphenidate the cut-off time (FDA: 3 h fasting, 4 h fed) is based on PD indeed (at that time ~90% of patients show the maximum effect). Makes sense. The EMA it its eternal wisdom asks to set the cut-off based on PK (a trough between the two parts). Splendid. Some subjects show just a shoulder (see there) and mean curves of the innovator are completely useless.

» AUMC (first order momentum, the area under the curve 'Concentration*time-time'). If we'll look at the physics analogy - the first-order momentum for the plain figure is the static momentum which defines the center of mass coordinates, while the second-order momentum is moment of inertia. The definition could be generalised to the n-th
» order momentum (why to limit ourselves by first order? There could be second or third order as well...).

Not only physics. Statistical distributions have also moments and we can interpret the behavior of drug molecules as a stochastic process. I love moments. I general

$$S_i=\int x^i\cdot f(x)dx$$

and in PK $$\small{i=0\ldots2}$$. Hence,

$$S_0=\int x^0\cdot f(x)dx = \int f(x)dx$$,
$$S_1=\int x^1\cdot f(x)dx = \int x\cdot f(x)dx$$,
$$S_2=\int x^2\cdot f(x)dx$$,

where in PK $$\small{x=t}$$ and $$\small{f(x)=C}$$. Then $$\small{AUC=S_0}$$, $$\small{AUMC=S_1}$$ and $$\small{MRT=AUMC/AUC}$$. $$\small{S_2}$$ is practically useless. OK, some people calculated $$\small{VRT = S_2/S_0-(S_1/S_0)^2}$$, the “Variance of Residence Times” or “Gravity Duration” (stop searching; out of fashion for decades). The coordinates $$\small{\{MRT\:|\:VRT\}}$$ define the “Center of Gravity” of the curve. Only nice to print a profile, cut it a out, push a pin through it, and make a weird whizz wheel for kids.

» MRT is a most common PK parameters that compels to look at the AUMC. There existed approaches4 to use MRT instead of T1/2 in order to choose the appropriate washout period.

Not only that. As a rule of thumb at $$\small{MRT}$$ ~⅔ of the drug is eliminated. It is very useful comparing PK models with different compartments. The slowest t½ might be misleading (see there, slides 24–28). There is a big problem with it. To get a reliable estimate of AUC one has to cover 95% (!) of AUC0–∞ (note that I’m not taking about BE but hard-core PK). For AUMC is should be 99%. I’m quoting Les Benet. Don’t blame me.

More to come.

Dif-tor heh smusma 🖖
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes
Astea
★★

Russia,
2020-06-13 10:17
(796 d 12:27 ago)

@ Helmut
Posting: # 21535
Views: 4,684

## Rattleback

Dear Helmut!

» Not only that (it’s a side-effect). What are we interested in? Extent of absoption (cleary AUC) and rate of absorption (ka and possibly tlag). ka is not easily accessible in NCA. Cmax is a composite surrogate (because influenced by AUC). Easy to show: Define any PK model and vary ƒ whilst keeping ka and tlag constant. Cmax will change… Cmax/AUC is an attempt to deal with that.

So what are the main requirements for the ideal PK metric? Not to change and have low variability? Returning to Cmax/AUC - do we need more strict conditions to prove BE on its base keeping in mind its much lower variability (narrow CI for example) - cause it's always within the limits? (In old (e.g. below 2013) russian protocols I even saw the opposite situation when the confidence limits for Cmax/AUC (as well as Cmax) were choosen to be "75,00-133,00%")

» Seriously, László (The Younger) asked me the same question years ago, which I could not answer. Martin helped us out. It doesn’t matter: The sum/difference of two normal distributions will be normal, the same here: It will be log-normal.

Much easily I should have read something about it somewhere on the forum. Can you please also clarify the distribution of T1/2: it is always presented as Mean±SD, is it correct? (Mean=Arithmetic Mean). It seemed to me I've read somewhere on the forum that for study planning we should use not the mean T1/2 but the confidence level for it, but I couldn't find this thread now. Is it correct to use standard approach to calculate CI for untransformed T1/2 as normally distributed data or should we use other approaches like nonparametric median confidence intervals or bootstrap?

» The jury is out. E.g., for biphasic methylphenidate the cut-off time (FDA: 3 h fasting, 4 h fed) is based on PD indeed (at that time ~90% of patients show the maximum effect). Makes sense.

That's correct but too much individual approach - we couldn't know beforehand all the thin nuances of the PD of the specific drug, sometimes it is even impossible to find in literature any PK data for some drugs while planning new study... So I think the overall approach should be more general.

» OK, some people calculated $$\small{VRT = S_2/S_0-(S_1/S_0)^2}$$, the “Variance of Residence Times” or “Gravity Duration” (stop searching; out of fashion for decades). The coordinates $$\small{\{MRT\:|\:VRT\}}$$ define the “Center of Gravity” of the curve.

Why not? The static moments of the plain curve 'Concentration-time' (C(t)) are:

$$M_t=\int t\cdot C(t)dt$$;
$$M_C=1/2\cdot\int C(t)^2dt$$

First is connected with MRT (muliplyed by total square of the curve), and the second should reflect some ideal Concentration-unit parameter (the height of the gravity center). Why not to use it like some alternative to Cmax?

» Only nice to print a profile, cut it a out, push a pin through it, and make a weird whizz wheel for kids.

Oh, if we'll proceed further in the power of moments we should be able to calculate moment of inertia - then one can take that curve to the orbital station in order to research Dzhanibekov effect.

» There is a big problem with it. To get a reliable estimate of AUC one has to cover 95% (!) of AUC0–∞ (note that I’m not taking about BE but hard-core PK). For AUMC is should be 99%. I’m quoting Les Benet. Don’t blame me.

Could you please explain this more detaily? What do you mean in getting a reliable estimate of AUC?

"Being in minority, even a minority of one, did not make you mad"
Helmut
★★★

Vienna, Austria,
2020-06-13 12:28
(796 d 10:16 ago)

@ Astea
Posting: # 21536
Views: 4,727

## Chatter

Hi Nastia!

» So what are the main requirements for the ideal PK metric?

You are asking a nasty question – rattling the foundations of BE by NCA. I like it.

» Not to change and have low variability?

IMHO, it should be selective, i.e., reflect changes of what we are interested in. For the extent of absorption it’s clearly AUC0–x. It’s another question what the x is… As we have seen in the other thread starting there, we get an unbiased estimate of T/R with AUC0–∞ and AUC0–t(common). How early could we stop sampling (x = ?) to get an unbiased estimate is – as you know – on my to-do list. I still hold that for IR products it’s much earlier than regulators think.
For the rate of absorption? Well, cough. Since Cmax is a composite metric, it can’t be selective for ka. We don’t have to re-invent the wheel; reading educates.1–6 Though I know >50% of the authors personally, I’m not biased.
Interesting the abstract6

Results: The outcome of a bioequivalence trial was shown to depend on the measure. Cmax/AUC reflected changes in ka, but not in F. AUC showed dependence on F, but virtually no dependence on ka. For Cmax, a 3- to 4-fold increase in ka and a concomitant 20% decrease in F, as well as corresponding changes in the opposite directions, resulted in bioequivalent outcomes.

But then

Conclusions: It was concluded that use of Cmax/AUC should be discouraged and that defining bioequivalence in terms of rate and extent of absorption has major problems.

(my emphasis)
IMHO, the conclusion contradicts the results. How come? Only because two authors were from the FDA and they didn’t want to change the rules? Why am I not surprised?
A picture tells more than a thousand words. A funny one of the paper:

A Cmax/AUC, B AUC, C Cmax

With $$\small{\frac{k_{a,T}}{k_{a,R}}=5\,\wedge \frac{F_T}{F_R}=0.8}$$ one has a high chance of passing Cmax (given, will fail on AUC). OK but as stated in the abstract there are combinations where ka and F are clearly different though both Cmax and AUC will pass. Bravo, well done! Great metric.
Reminds me on another paper7 by authors of the FDA assessing the performance of AUC0–t and AUC0–∞. Results: T/R-ratios very similar, AUC0–∞ more variable. The FDA’s consequences: Use both.

» Returning to Cmax/AUC - do we need more strict conditions to prove BE on its base keeping in mind its much lower variability (narrow CI for example) - cause it's always within the limits?

Why? The conventional limits are based on the assumption that a of 20% is clinically not relevant. Leaving NTIDs aside, do we narrow the limits (say, for AUC) only because the variability of some drugs is low? Nope. If we would make the limits dependent on the variability, we would end up in reference-scaling chaos.

» (In old (e.g. below 2013) russian protocols I even saw the opposite situation when the confidence limits for Cmax/AUC…

I know. Strange.

» … (as well as Cmax) were choosen to be "75,00-133,00%")

I understand that. Before the European 2001 Note for Guidance, many products got even an approval with 30% (70.00–142.86%).

» Can you please also clarify the distribution of T1/2: it is always presented as Mean±SD, is it correct? (Mean=Arithmetic Mean).

You rub salt into my wounds. Let’s step back. Rate constants have a unit of 1/time. Hence, the correct location parameter is the harmonic mean. Its dispersion parameter is the jackknife standard deviation (in WinNonlin’s terminology: Pseudo SD). For t½ you have two options: Use the same as well (as I do though I’m not sure about the distribution; Γ?) or go with nonparametrics (x̃, quartiles).

» It seemed to me I've read somewhere on the forum that for study planning we should use not the mean T1/2 but the confidence level for it, but I couldn't find this thread now.

I’m too lazy to search as well. This one about extremes?

» Is it correct to use standard approach to calculate CI for untransformed T1/2 as normally distributed data …

Anything is better than the mean (see this presentation, slides 64–66).

» … or should we use other approaches like nonparametric median confidence intervals or bootstrap?

Sounds good though “nonparametric CI” is a little bit strange. I think that I once saw a paper about it, not sure. However, if you don’t have data of a previous study… What you find in the public domain is often x±SD or min/max.

» » E.g., for biphasic methylphenidate the cut-off time (FDA: 3 h fasting, 4 h fed) is based on PD indeed (at that time ~90% of patients show the maximum effect). […]
»
» That's correct but too much individual approach - we couldn't know beforehand all the thin nuances of the PD of the specific drug, sometimes it is even impossible to find in literature any PK data for some drugs while planning new study... So I think the overall approach should be more general.

Early and late partial AUCs are only relevant for multiphasic MR products. Luckily I met only three so far: zolpidem, methylphenidate, amphetamine(s). For the first two there is a PD based justification possible. For the last one – no idea.

» Why not? The static moments of the plain curve 'Concentration-time' (C(t)) are:
»
»

$$M_t=\int t\cdot C(t)dt$$;
» $$M_C=1/2\cdot\int C(t)^2dt$$

» First is connected with MRT (muliplyed by total square of the curve), and the second should reflect some ideal Concentration-unit parameter (the height of the gravity center). Why not to use it like some alternative to Cmax?

Back in the days when I used my own software, I reported all it could calculate (hey, look how clever I am). Arrogant attitude. Confused sponsors and regulators. What I learned: The variability of VRT sucks. Not surprising cause we have $$\small{t^2}$$ and $$\small{C^2}$$ in it.

» » Only nice to print a profile, cut it a out, push a pin through it, and make a weird whizz wheel for kids.
»
» Oh, if we'll proceed further in the power of moments we should be able to calculate moment of inertia - then one can take that curve to the orbital station in order to research Dzhanibekov effect

Didn’t know that one! BTW, the center of gravity can be outside of the profile (scroll down in this post). Aboriginals know that for ages.

» » To get a reliable estimate of AUC one has to cover 95% (!) of AUC0–∞ (note that I’m not taking about BE but hard-core PK). For AUMC is should be 99%. I’m quoting Les Benet. Don’t blame me.
»
» Could you please explain this more detaily? What do you mean in getting a reliable estimate of AUC?

I was talking about PK modeling. IIRC, Les Benet said that at the BioInternational in Munich 1994. Not sure whether it’s mentioned in the book.

1. Endrényi L, Fritsch S, Yan W. Cmax/AUC Is a Clearer Measure Than Cmax for Absorption Rates in Investigations of Bioequivalence. Int J Clin Pharm Ther Toxicol. 1991;29(10):394–9. PMID 1748540.
2. Schall R, Luus HG. Comparison of absorption rates in bioequivalence studies of immediate release drug formulations. Int J Clin Pharmacol Ther. 1992;30(5):153–9. PMID 1592542.
3. Endrényi L, Yan W. Variation of Cmax and Cmax/AUC in investigations of bioequivalence. Int J Clin Pharm Ther Toxicol. 1993;31(4):184–9. PMID 8500920.
4. Lacey LF, Keene ON, Duquesnoy C, Bye A. Evaluation of Different Indirect Measures of Rate of Drug Absorption in Comparative Pharmacokinetic Studies. J Pharm Sci. 1994;83(2):212–5. doi:10.1002/jps.2600830219.
5. Schall R, Luus HG, Steinijans VW, Hauschke D. Choice of Characteristics and Their Bioequivalence Ranges for the Comparison of Absorption Rates of Immediate-Release Drug Formulations. Int J Clin Pharmacol Ther. 1994;32(7):323–8. PMID 7952792.
6. Tozer TN, Bois FY, Hauck WW, Chen M-L, Williams RL. Absorption Rate Vs. Exposure: Which Is More Useful for Bioequivalence Testing? Pharm Res. 1996;13(3):453–56. doi:10.1023/a:1016061013606.
7. Bois FY, Tozer TN, Hauck WW, Chen M-L, Patnaik R, Williams RL. Bioequivalence: Performance of Several Measures of Extent of Absorption. Pharm Res. 1994;11(5):715–22. doi:10.1023/A:1018932430733.

My post № 5,000.

Edit: I explored my data. Drug X, 5–60 mg (linear PK proven), same bioanalytical method (enantio­selective stable isotope IS GC/MS, LLOQ dependent on the dose), sampling for 24 h, 3–7 time points for the estimation of λz, extrapolated AUC <10%.

I was wrong for many years. Seems that I have to revise my procedures and go with the median or geometric mean in the future.

Dif-tor heh smusma 🖖
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes
Astea
★★

Russia,
2020-06-16 01:33
(793 d 21:12 ago)

@ Helmut
Posting: # 21540
Views: 4,379

## Reinventing the Hula-Hoop

Dear Friends!

I don't quite keep up with all directions of thought, but I am extremely grateful for your profound answers! I will discover your explanations and links step by step.

Dear Helmut!
You adviced me not to google VRT, but how could I avoid this temptation?
Now I think that you are slightly confused: VRT do not define the center of gravity. According to J.E. Riviere1:

$$VRT=[\int (t-MRT)^2\cdot C(t)dt]/AUC$$.

It has t2 under the integration, so it is just what we consider as an analogue of moment of inertia, that is the second moment, besides its dimension is 'time2'.

VRT is used to calculate the coefficient of variation of residence times (CVRT):

$$CVRT=\sqrt (VRT)/MRT$$,

which is a dimensionless parameter that provides 'the dynamics and heterogeneity of drug distribution'.

On the other hand the center of gravity would be defined by the following equation:

$$C_c=1/2\cdot[\int C(t)^2\cdot dt]/AUC$$.

It has the dimension of concentration. It was suggested by Lassen and Perl in 1979 to use it as a perfect parameter that is sensitive to both the rate and the extent of absorption, which is not the case for the AUMC/AUC measure.

That same feeling when you invent something that was already invented long before you was born (considering the center of gravity of hula-hoop - links to the concept of solipsism)

» Only nice to print a profile, cut it a out, push a pin through it, and make a weird whizz wheel for kids.

You've already discussed it here (thanks to mittyri I've finally found that post and finally got rid of deja vu).
Obviously as you've mentioned C2 will give us much variability, but may be it is not worth to throw momentally this parameter to the bin. It reflects the extent of absorbtion and could reflect the major properties of the curve even in the case when it is outside it. Not the case of MDT. Returning to my initial question: what do you think about calculating MDT for multiphase profiles?

Returning to MRT: two similar parameters should be mentioned:
MAT Mean Absorbtion Time (MAT=MRTni-MRTIV, where ni is any noninstanteneous administration) and
MTT Mean Transit Time2.
[1] Riviere, J. E. Comparative Pharmacokinetics Principles, Techniques, and Applications, DOI:10.1002/9780470959916
[2] Veng - Pedersen, P. 1989a. Mean time parameters in pharmacokinetics: defi nition, computation and clinical implications (part I). Clinical Pharmacokinetics. 17 : 345 – 366.
[3] Veng - Pedersen, P. 1989b. Mean time parameters in pharmacokinetics: defi nition, computation and clinical implications (part II). Clinical Pharmacokinetics. 17 : 424 – 440.

"Being in minority, even a minority of one, did not make you mad"
Helmut
★★★

Vienna, Austria,
2020-06-16 13:07
(793 d 09:37 ago)

@ Astea
Posting: # 21543
Views: 4,343

## Hula-Hoop references

Hi Nastia,

Sadistically I throw some more into the arena. A nice quote in Brockmeier’s review 19

In 1958 [sic], F.H. Dost defined the mean life-span (“mittlere Lebensdauer”) of a total number of N molecules as the arithmetic mean of all times “zi” of any one of the N molecules residing in a pharmacokinetic system. This pharmacokinetic characteristic did not attract special interest for several years.

(my emphases)
BTW, Friedrich Hartmut Dost termed “Pharmakokinetik” in 1953 1

Pharmakokinetik ist die Lehre von der quantitativen Auseinandersetzung zwischen Organismus und einverleibten Pharmakon, sonst nichts weiter.
Pharmacokinetics is the science of the quantitative relationship between the organism and incorporated drug, nothing else. (my translation)

Pharmakokinetik is a portmanteau word from ancient Greek φαρμακός (drug) + κινητικός (putting in motion). In the early 1960s (by whom: Gerald Levy, Sidney Riegelman, John Wagner, Milo Gibaldi?) it was translated to pharmacokinetics.

Picky in the list of abbreviations 19

AUC = Area under the curve, most frequently computed by the trapezoidal rule and therefore more appropriately denoted as area under the data (AUD).
AUMC = Area under the first moment curve, i.e. the curve derived by multiplying the concentration by time (C(t)×t). Most frequently computed with the data and integrated by the trapezoidal rule and therefore more appropriately denoted as area under the first moment data (AUMD).

(my emphases)

» » There is a big problem with it. To get a reliable estimate of AUC one has to cover 95% (!) of AUC0–∞ (note that I’m not taking about BE but hard-core PK). For AUMC is should be 99%. I’m quoting Les Benet. Don’t blame me.
»
» Could you please explain this more detaily? What do you mean in getting a reliable estimate of AUC?

I was wrong (not for the first time). 3
$$C_p=\frac{A}{k_a-k_e}(\textrm{e}^{-k_et}-\textrm{e}^{-k_at}) \tag{6}$$

When the time course is measured until the plasma concentration becomes 5% of its maximum, the relative cutoff errors in AUC, MRT, and VRT are smaller than 5%, 10%, and 40%, respectively, and they are independent of the value $$\small{A}$$ in equation 6. If the time course is measured up to the time when plasma concentration becomes 1% of its maximum, the relative errors in AUC, MRT, and VRT are smaller than about 1%, 2%, and 10%, respectively.

1. Dost FH. Der Blutspiegel. Konzentrationsabläufe in der Kreislaufflüssigkeit. Leipzig: VEB Thieme; 1953.
2. Dost FH. Über ein einfaches statistisches Dosis-Umsatz-Gesetz. Klin Wochenschr. 1958; 36(14):655–7. doi:10.1007/bf01488743.
3. Yamaoka K, Nakagawa T, Uno T. Statistical Moments in Pharmacokinetics. J Pharmacokin Biopharm. 1978;6;547–58. doi:10.1007/BF01062109.
4. Cutler DJ. Theory of the mean absorption time, an adjunct to conventional bioavailability studies. J Pharm Pharmacol. 1978;30(8):476–8. doi:10.1111/j.2042-7158.1978.tb13296.x.
5. von Hattingberg HM, Brockmeier D. Standardisierung von Rechenmodellen zur Prüfung der Bioverfügbarkeit von Arzneimitteln. In: Rietbrock N, Schnieders B, editors. Bioverfügbarkeit von Arzneimitteln. Stuttgart, New York: Gustav Fischer; 1979. p. 191–209. ISBN 3-437-10654-6.
6. Riegelman S, Collier P. The application of statistical moment theory to the evaluation of in vivo dissolution time and absorption time. J Pharmacokin Biopharm. 1980;8:509–34. doi:10.1007/BF01059549.
7. Weiss M. Residence Time and Accumulation of Drugs in the Body. Int J Clin Pharm Ther Toxicol. 1981;19(2):82–5. PMID 7216555.
8. Hattingberg HM, Brockmeier D. A Concept for the Assessment of Bioavailability in Complex Systems in Terms of Amounts and Rates. In: Botzler G, van Rossum JM, editors. Pharmacokinetics During Drug Development: Data Analysis and Evaluation Techniques. Stuttgart, New York: Gustav Fischer; 1982. p. 315–23. ISBN 3-437-10654-6.
9. Chan KK, Gibaldi M. Estimation of Statistical Moments and Steady-State Volume of Distribution for a Drug Given by Intravenous Infusion. J Pharmacokin Biopharm. 1982;10(5):551–8. doi:10.1007/BF01059037.
10. Gouyette A. Pharmacokinetics: Statistical Moment Calculations. Drug Res. 1983;33(1):173–6. PMID 6681972.
11. Matis JH, Wehrly TE, Metzler CM. On Some Stochastic Formulations and Related Statistical Moments of Pharmacokinetic Models. J Pharmacokin Biopharm. 1983;11(1);77–92. doi:10.1007/BF01061769.
12. Beal SL. Some Clarifications Regarding Moments of Residence Times With Pharmacokinetic Models. J Pharmacokin Biopharm. 1987;15(1):75–92. doi:10.1007/BF01062940.
13. Brockmeier D, von Hattingberg HM. Mean Residence Time. Methods Find Exp Clin Pharmacol. 1986;8(5):309–12. PMID 3724305.
14. Kasuya Y, Hirayama H, Kubota N, Pang KS. Interpretation and Estimates of Mean Residence Time With Statistical Moment Theory . Biopharm Drug Dispos. 1987;8(3):223–34. doi:10.1002/bdd.2510080304.
15. Nakashima E, Benet LZ. General treatment of mean residence time, clearance, and volume parameters in linear mammillary models with elimination from any compartment. J Pharmacokin Biopharm. 1988;16(5):475–92. doi:10.1007/BF01062381.
16. Purves RD. Optimum Numerical Integration Methods for Estimation of Area-Under-the-Curve (AUC) and Area-under-the-Moment-Curve (AUMC). J Pharmacokin Biopharm. 1992;20(3):211–26. doi:10.1007/BF01062525.
17. Cheng H, Gillespie WR, Jusko WJ. Mean Residence Time Concepts for Non-Linear Pharmacokinetic Systems. Biopharm Drug Disp. 1994;15:627–41. doi:10.1002/bdd.2510150802.
18. Cheng H, Gillespie WR. Volumes of distribution and mean residence time of drugs with linear tissue distribution and binding and nonlinear protein binding. J Pharmacokin Biopharm. 1996;24(4):389–402. doi:10.1007/bf02353519.
19. Brockmeier D. Mean Time Concept and Component Analysis in Pharmacokinetics. Int J Clin Pharmacol Ther. 1999;37(11):555–51. PMID 10584977.

Dif-tor heh smusma 🖖
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes
mittyri
★★

Russia,
2020-06-17 14:28
(792 d 08:16 ago)

@ Helmut
Posting: # 21547
Views: 4,196

## Prof. Keller vs. Yamaoka

Hi Helmut,

» I was wrong (not for the first time). 1
» $$C_p=\frac{A}{k_a-k_e}(\textrm{e}^{-k_et}-\textrm{e}^{-k_at}) \tag{6}$$
» When the time course is measured until the plasma concentration becomes 5% of its maximum, the relative cutoff errors in AUC, MRT, and VRT are smaller than 5%, 10%, and 40%, respectively, and they are independent of the value $$\small{A}$$ in equation 6. If the time course is measured up to the time when plasma concentration becomes 1% of its maximum, the relative errors in AUC, MRT, and VRT are smaller than about 1%, 2%, and 10%, respectively.

Prof. Keller disagrees 2:
<...>for an acceptable estimate of the MRTtrunc the last concentration to be measured should be 1.096 % of the initial concentration value or less. Accordingly, the concentration curve must be followed for approximately a 2 log decline (10–2*C0 <= Cn). Thus, the truncation at a concentration that is 1 % of the initial concentration will result in a 5 % error of the cut MRT estimate and not 2 % as stated by Yamaoka et al. According to mono-exponential kinetics, the near 2 log period lasts for 6.51 times the elimination half-life<...>

By the way I couldn't follow Yamaoka's logic regarding that magic cut-off errors. How did they find it?
Regarding prof.Keller's note: he's using IV data for cut-off errors estimation, so it is not clear why he did compare it with 1-cpt model with oral first-order absorption in Yamaoka's article.

1. Yamaoka K, Nakagawa T, Uno T. Statistical Moments in Pharmacokinetics. J Pharmacokin Biopharm. 1978;6;547–58. doi:10.1007/BF01062109.
2. Keller F, Hartmann B, Czock D. Mean residence time as estimated from cropped and truncated moments. Arzneimittelforschung. 2009;59(7):377-381. doi:10.1055/s-0031-1296411.

Kind regards,
Mittyri
Astea
★★

Russia,
2020-06-23 14:41
(786 d 08:03 ago)

@ mittyri
Posting: # 21565
Views: 3,763

## SHAM(e) math

Dear Helmut!

» Reach for the stars, even if you have to stand on a cactus. Susan Longacre
Uno puede estar mirando las estrellas y al mismo tiempo verse la punta de las pestañas (Julio Cortázar)

I am very grateful for your answers and the references provided! I've found some of them and will search for others although for some reason I have strong doubts that our local library has books on pharmacokinetics on german printed in 50th

» Not only that. As a rule of thumb at $$\small{MRT}$$ ~⅔ of the drug is eliminated. It is very useful comparing PK models with different compartments. The slowest t½ might be misleading (see there, slides 24–28). There is a big problem with it. To get a reliable estimate of AUC one has to cover 95% (!) of AUC0–∞ (note that I’m not taking about BE but hard-core PK). For AUMC is should be 99%. I’m quoting Les Benet. Don’t blame me.

I was wondering from where such a rule of thumb was going and integrated the area for simple exponential elimination. It turns out that at MRT (1-exp(-1))~0,632 of the drug is eliminated for IV and slightly lower for EV (so the rule of pinky is 0,632 versus the rule of thumb (2/3=0,(6))

As for physics there exists inaccuracy in the considerations on the slide "Excursion to Hydrodynamics" . "Same proportions is emptied in the same time interval" is true only when you are solving school problems with a pool. Exactly the unexpired volume leaked depends on the form of the vessel. For the cylindric vessel for example water height and thus the volume is proportional to t2. If you want to have a constant proportion you need a vessel with a form of parabola x4 that is clepsydra or consider Mariotte’s bottle.

» What I learned: The variability of VRT sucks. Not surprising cause we have $$\small{t^2}$$ and $$\small{C^2}$$ in it.

I've calculated Cc for several real studies according to simple linear trapezoidal rule:
$$C_c=\frac{1}{3}\frac{\sum\limits_{i}(t_{i+1}-t_i)(C^{2}_{i}+C^{2}_{i+1}+C_{i}\cdot C_{i+1})}{\sum\limits_{i}(t_{i+1}-t_{i})(C_{i+1}+C_i)}\tag{7}.$$ Although it has C2 in it, it's variability was always lower than that of Cmax, but I should've check it more carefully.

Dear ElMaestro!

» I think F may be in its own right also included on your list of crackpot ideas from the odd sock drawer? PMDA have a sentence about it in their guidance. "If F can be calculated by deconvolution, F may be used instead of AUC"

Thank you! I will definitely add it to my collection of weird PK parameters! Need to know more about deconvolution...

Dear mittyri!

» By the way I couldn't follow Yamaoka's logic regarding that magic cut-off errors. How did they find it?

I am puzzled with the same question. How did they calculated the time to reach 5% of Cmax?
I slightly modified the Helmut's considerations on the article of Scheerans et al. (2008)
Let us consider a one-compartment model with first-order absorbtion of the form:
$$C=\frac{A}{k_a-k_e}(\textrm{e}^{-k_et}-\textrm{e}^{-k_at}) \tag{6}$$
then residual area (1-AUC0-t/AUC0-inf) should be as follows:
$$AUC_{resid}(x,t)=\frac{x\textrm{e}^{-t\cdot k_e}-\textrm{e}^{-x\cdot k_e*t}}{x-1},\quad \textrm{where}\qquad x=\frac{k_a}{k_e}.$$ Let n define the ratio of t to T1/2,e, then
$$AUC_{resid}(x,n)=\frac{x\cdot2^{-n}-2^{-nx}}{x-1}\sim \frac{x\cdot 2^{-n}}{x-1}\qquad for\qquad nx>>1. \tag{8}$$
In order to estimate the duration of sampling to achieve specific AUCresid we can use the simplifyed formula
$$n=\textrm{log}_2\left(\frac{x}{(x-1)AUC_{resid}}\right) \tag{9},$$
for example for x=2 and AUCresid=1% the duration should be n=7.64 T1/2, for AUCresid=20% the duration should be n=3.32 T1/2 (the exact value is 3.24.)
In particular, $$AUC_{resid}(T_{1/2},x)=\frac{x-2^{1-x}}{2(x-1)};\quad AUC_{resid}(T_{max},x)=\frac{x^{\frac{2-x}{1-x}}-x^{\frac{x}{1-x}}}{(x-1)};\quad AUC_{resid}(2T_{max},x)=\frac{x^{\frac{3-x}{1-x}}-x^{\frac{2x}{1-x}}}{(x-1)}.$$
AUCresid(Tmax,x) is a monotone function of x limited from 2/e (0.736) to 1;
AUCresid(2Tmax,x) is a monotone function of x limited from 3/e3 (0.406) to 1.

"Being in minority, even a minority of one, did not make you mad"
Helmut
★★★

Vienna, Austria,
2020-06-23 15:55
(786 d 06:49 ago)

@ Astea
Posting: # 21566
Views: 3,737

## SHAM(e) math

Hi Nastia,

» Uno puede estar mirando las estrellas y al mismo tiempo verse la punta de las pestañas (Julio Cortázar)

Though I never just couldn’t get into Cortázar’s books, that’s a nice quote (though having both estrellas and pestañas in focus would be a difficult feat).

» […] I have strong doubts that our local library has books on pharmacokinetics on german printed in 50th

I believe it. I had only the “expanded edition”:

Dost FH. Grundlagen der Pharmakokinetik. Stuttgart: Verlag G. Thieme; 1968.

Forget to steal it when I left my CRO.

» » As a rule of thumb at $$\small{MRT}$$ ~⅔ of the drug is eliminated. […]
» I was wondering from where such a rule of thumb was going and integrated the area for simple exponential elimination. It turns out that at MRT (1-exp(-1))~0,632 of the drug is eliminated for IV and slightly lower for EV (so the rule of pinky is 0,632 versus the rule of thumb (2/3=0,(6))

Absolutely correct! This was a presentation for physicians (‼); I wanted to keep it simple. A relative error of 5.2% doesn’t hurt to make a point. Of course, much worse than Archi­medes’ phantastic $$\small{3+\frac{10}{71}}<\pi<3+\frac{1}{7}$$.

» As for physics there exists inaccuracy in the considerations on the slide "Excursion to Hydrodynamics". "Same proportions is emptied in the same time interval" is true only when you are solving school problems with a pool. Exactly the unexpired volume leaked depends on the form of the vessel. For the cylindric vessel for example water height and thus the volume is proportional to t2. If you want to have a constant proportion you need a vessel with a form of parabola x4 that is clepsydraor consider Mariotte’s bottle.

Correct again! I brainlessly used examples of old textbooks (as usual). Homework: what happens if we drill a hole in a Klein bottle?

» I've calculated Cc for several real studies according to simple linear trapezoidal rule:
» $$C_c=\frac{1}{3}\frac{\sum\limits_{i}(t_{i+1}-t_i)(C^{2}_{i}+C^{2}_{i+1}+C_{i}\cdot C_{i+1})}{\sum\limits_{i}(t_{i+1}-t_{i})(C_{i+1}+C_i)}$$ Although it has C2 in it, it's variability was always lower than that of Cmax, but I should've check it more carefully.

Surprises me. Given, I didn’t assess it for ages. Maybe I’m wrong again.

Dif-tor heh smusma 🖖
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes
Astea
★★

Russia,
2020-06-23 21:41
(786 d 01:04 ago)

@ Helmut
Posting: # 21569
Views: 3,683

## OT: Möbius strip

Dear Helmut!

» Homework: what happens if we drill a hole in a Klein bottle?

Hey, how did you know that I've got a model of it?
Unfortunatelly I am not good at topology and do not know what will happen. But I know the answers for the questions what do we need to drill a square hole and also what profile one should use to ride a bike with square wheels

"Being in minority, even a minority of one, did not make you mad"
Helmut
★★★

Vienna, Austria,
2020-06-24 11:29
(785 d 11:16 ago)

@ Astea
Posting: # 21572
Views: 3,628

## OT: Möbius strip

Hi Nastia,

» Hey, how did you know that I've got a model of it?

I didn’t know but I’m not surprised.

» Unfortunatelly I am not good at topology…

cup → bagel → cup → …

Welcome to the club. The worst is knot theory – I have books about it. Headaches guaranteed.
For sure you know what happens if we cut a Möbius strip once. Do you know what happens if we cut it twice? Makes a great party joke.

» But I know the answers for the questions what do we need to drill a square hole…

Punch, OK. But drill?

» … and also what profile one should use to ride a bike with square wheels

So do I.

https://www.r-bloggers.com/topological-tomfoolery-in-r-plotting-a-mobius-strip
library(rgl)
library(plot3D)
###############
# Möbius band #
###############

R <- 5
u <- seq(0, 2 * pi, length.out = 100)
v <- seq(-1, 1, length.out = 100)
m <- mesh(u, v)
u <- m$x v <- m$y
x <- (R + v/2 * cos(u /2)) * cos(u)
y <- (R + v/2 * cos(u /2)) * sin(u)
z <- v/2 * sin(u / 2)
invisible(open3d())
bg3d(color = "#F3F3F3")
surface3d(x, y, z, color= "#87CEFA")
title3d(main = "Moebius band")
# Rotate with mouse-left, wheel to zoom.
################
# Klein bottle #
################

u <- seq(0, pi, length.out = 100)
v <- seq(0, 2 * pi, length.out = 100)
m <- mesh(u, v)
u <- m$x v <- m$y
x <- (-2 / 15) * cos(u) * (3 * cos(v) - 30 * sin(u) + 90 * cos(u)^4 * sin(u) -
60 * cos(u)^6 * sin(u) + 5 * cos(u) * cos(v) * sin(u))
y <- (-1 / 15) * sin(u) * (3 * cos(v) - 3 * cos(u)^2 * cos(v) - 48 *
cos(u)^4 * cos(v) + 48 * cos(u)^6 * cos(v) - 60 * sin(u) + 5 * cos(u) *
cos(v) * sin(u) - 5 * cos(u)^3 * cos(v) * sin(u) - 80 * cos(u)^5 *
cos(v) * sin(u) + 80 * cos(u)^7 * cos(v) * sin(u))
z <- (+2 / 15) * (3 + 5 * cos(u) * sin(u)) * sin(v)
invisible(open3d())
bg3d(color = "#F3F3F3")
surface3d(x, y, z, color= "#87CEFA", alpha = 0.5)
title3d(main = "Klein bottle")

Dif-tor heh smusma 🖖
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes
Helmut
★★★

Vienna, Austria,
2020-06-13 15:28
(796 d 07:17 ago)

@ Astea
Posting: # 21537
Views: 4,647

## More stuff

Hi Nastia,

another part. I love your attitudes of questioning the foundations of what we are doing. Proves that you are a true scientist.

Reach for the stars,
even if you have to stand on a cactus.
Susan Longacre

I know that, been there. But you do it barefooted. Kudos!

» Capical was invented in order to explore the shape of the peak more precisely. As far as I understood it was first mentioned in an article in 19885 […] If Capical is so promising parameter, why can't I find any trials with it in results (excepting [6] and a paper on controlled release gastroretentive dosage forms tested on dogs). Or my search methods are too poor? I would be grateful if anyone will give an example of it's calculation.

That’s the first reference I’m aware of as well. Never tried it.

» The concentration at the end of the intended dosing interval (Cτ) should serve as an analogue of Cmin with lack of multiple dose studies.

Yes, but that’s a sad, sad story. I did my best at the GBHI in Amsterdam (2018) as well as Nuno Silva and Jack Cook in Bethesda (2019). Voices crying in the wilderness.

» The lag time matters when we deal with delayed-release formulations. Note that Phoenix defines it as time of observation prior to the first observation with a measurable (non-zero) concentration but not the time to the first observation with a measurable concentration. There could also be other ways7 to define it.

Correct. Lag-time means before a concentrations is measurable, hence, it cannot be the first measured one. It’s somewhere before the first measured one. As we discussed in the linked thread, I think that Detlew’s approach is pragmatic. Ref.7 would smell too much of modeling for regulators, I guess.

» T50%between (half-value duration)…

AFAIK, the HVD (Half Value Duration) appeared for the first time already in 19741 as $$\small{\Delta_{1/2}}$$ – part of the “Retardquotient” which compares “the degree of retardation” with IR as $$\small{R_\Delta=\frac{\Delta_{1/2,MR}}{\Delta_{1/2,IR}}}$$. Suggested was
$$\small{\begin{array}{ll} R_\Delta\leqslant 1 & \text{no retardation} \\ R_\Delta\sim 1.5 & \text{weak retardation} \\ R_\Delta\sim 2 & \text{medium retardation} \\ R_\Delta\geqslant 3 & \text{strong retardation} \\ \end{array}}$$

I found it in my bible2 (stop searching; in German and out of print for ages) and some later stuff.3,4 The Two Lászlós renamed it to HaVD to avoid confusion with highly variable drugs.
Enduring the discussions about PK metrics at the GBHI-workshops was like a flash-back after a bad LSD-trip. Stirring up murky waters which cleared more than thirty years ago.

» "Therapeutic response" ('Where it exists, consideration must be given to the "therapeutic window."')8. But aren't there limitations that could not afford to make a statement of any correlation between plasma concentrations and effect?

You are a keen thinker! Though Jerome was head of the FDA’s CDER at that time, he felt into the trap of confusing PK with PD.

Pharmacokinetics may be simply defined as
what the body does to the drug,
as opposed to pharmacodynamics, which may be defined as
what the drug does to the body.
Leslie Z. Benet

PD is connected via a more or less complicated link model to PK. Ask a modeler for horror stories. One is sure: The time point of Emax is later than tmax and differences between formulations might be dampened in PD (though the variability of PD measurements can be awful). Ask ten physicians and eleven will tell you that safety is directly related to Cmax. Not that easy. Of course the same holds for Cmin (or Cτ if you prefer) which is not directly related to efficacy. Hence the bracketing approach of comparing MR with IR (see this post) is – from a purely scientific perspective – built on sand. Unfortunately for  many  most drugs we have no clue about PK/PD. We have PK data from Phase I and PD data from Phase II. Quite often we have in Phase II only sparse sampling and the PK/PD models are not that good (pun!).

Another5 about $$\small{AUC\frac{above\, C_{av}}{below\, C_{av}}}$$ (sorry, in German except the abstract):

It goes back to a proposal of the late Harold Boxenbaum6 of the days when “the flatter, the better” was a common slogan. I liked Harold because he didn’t give a shit about dress codes (always showed up in jeans, leather jacketed, wearing cowboy boots and a Stetson) and had a sharp tongue.

1. Meier J, Nüesch E, Schmidt R. Pharmacokinetic criteria for the evaluation of retard formulations. Eur J Clin Pharmacol. 1974;7:429-32.
2. Meier J. Bioverfügbarkeit und Absorption. In: Meier J, Rettig H, Hess H, editors. Biopharmazie. Theorie und Praxis der Pharmakokinetik. Stuttgart, New York: Georg Thieme; 1981. p. 247–76. ISBN 3-13-603106-6.
3. Steinijans VW. Pharmacokinetic Characteristics of Controlled Release Products and Their Biostatistical Analysis. In: Gundert-Remy U, Möller H, editors. Oral Controlled Release Products – Therapeutic and Biopharmaceutic Assessment. Stuttgart: Wissenschaftliche Verlagsanstalt; 1988. p. 99–115.
4. Steinijans VW, Hauschke D. Modified-Release Dosage Forms: Acceptance Criteria and Statistics for Bioequivalence. Drug Inf J. 1993;27:903–9. doi:10.1177/009286159302700332.
5. Blume H, Siewert M, Steinijans V, Stricker H. Bioäquivalenz von per os applizierten Retard-Arzneimitteln. Konzeption der Studien und Entscheidung über Austauschbarkeit. Pharm Ind. 1989;51(9):1025–33.
6. Boxenbaum H. Pharmacokinetic Determinants in the Design and Evaluation of Sustained-Release Dosage Forms. Pharm Res. 1984;1(2):82–8. doi:10.1023/A:1016355431740.

Dif-tor heh smusma 🖖
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes
mittyri
★★

Russia,
2020-06-14 22:24
(795 d 00:20 ago)

@ Astea
Posting: # 21539
Views: 4,508

## MRT and Gravity duration

Dear Astea,

» MRT is a most common PK parameters that compels to look at the AUMC. There existed approaches4 to use MRT instead of T1/2 in order to choose the appropriate washout period.

Just my 2 cents regarding MRT. You cannot estimate it for non-IV ways of administration. For non-IV that parameter is called 'gravity duration', since includes both MIT and MRT. See here for discussion.

Kind regards,
Mittyri
ElMaestro
★★★

Denmark,
2020-06-16 09:46
(793 d 12:58 ago)

@ Astea
Posting: # 21541
Views: 4,346

## Fantastic PK parameters and where to find them

Hi Astea

» While reading the forum I sometimes face to the statements that other PK parameters should be used in future researches for SD studies. Below is my collection of stange or rare parameters. I would be grateful if you'll comment on their properties, details of calculation and the perspectives of its using.

I think F may be in its own right also included on your list of crackpot ideas from the odd sock drawer? PMDA have a sentence about it in their guidance. "If F can be calculated by deconvolution, F may be used instead of AUC"

» The distribution type of the function is also questionable (ratio of log-normal - paranormal ?).

Paranormal is exactly what it is.

Conveniently, ln(A)-ln(B)=ln(A/B). If A is normal and B is normal then their sum (difference) is normal, and it is trivial to work mean and variance out. But the ratio of two normal distributions is distinctly not normal. We can't just say the ratio of two log-normals is normal (or log-normal, depending on the level of liguistic nitpicking); we need to keep in mind on which scale we subtract or add, and on which scale we do the ratio.
Kindly send me a telegram when someone works out the distribution of the ratio of two normals

Pass or fail!
ElMaestro
mittyri
★★

Russia,
2020-06-16 10:54
(793 d 11:50 ago)

@ ElMaestro
Posting: # 21542
Views: 4,344

## Cauchy distribution

Hi ElMaestro,

» Kindly send me a telegram when someone works out the distribution of the ratio of two normals

Here ya go: Cauchy distribution

Kind regards,
Mittyri
Helmut
★★★

Vienna, Austria,
2020-06-16 13:14
(793 d 09:30 ago)

@ mittyri
Posting: # 21544
Views: 4,313

## Cauchy distribution

Hi mittyri,

the Cauchy distribution is a funny one. Undefined mean and variance.

Dif-tor heh smusma 🖖
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes
ElMaestro
★★★

Denmark,
2020-06-20 10:33
(789 d 12:12 ago)

@ mittyri
Posting: # 21551
Views: 4,002

## Cauchy distribution

Hi mittyri,

» Here ya go: Cauchy distribution

The Cauchy distribution is relevant to this question only to the extent that we can assume ln(T)=ln(R) when both have zero mean. It is thus a special case only, and therefore one that does not in practice correspond to the generally relevant question.
In perspective, history has been somewhat cruel to anyone assume T=R for example in relation to power and sampe sizes. In other words, what is needed (if there is a need at all?!?) is a "Cauchy distribution with the equivalent of a noncentrality parameter", for lack of better wording.

Pass or fail!
ElMaestro
mittyri
★★

Russia,
2020-06-20 23:04
(788 d 23:40 ago)

@ ElMaestro
Posting: # 21553
Views: 3,938

## noncentral normal ratio

Hi ElMaestro,

» In perspective, history has been somewhat cruel to anyone assume T=R for example in relation to power and sampe sizes. In other words, what is needed (if there is a need at all?!?) is a "Cauchy distribution with the equivalent of a noncentrality parameter", for lack of better wording.

but in the link above you can see
Uncorrelated noncentral normal ratio
...Under certain conditions, a normal approximation is possible...

Correlated noncentral normal ratio
...The offset result is also consistent with the "Cauchy" correlated zero-mean Gaussian ratio distribution in the first section...

Kind regards,
Mittyri