Astea ★★ Russia, 20200612 16:22 (1405 d 01:58 ago) Posting: # 21533 Views: 7,658 

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. C_{max}/AUC is referred sometimes to be more appropriate in BE studies than C_{max} cause it has much lower variability (here we discuss C_{max}/AUC_{last} but not C_{max}/AUC_{inf} which is worse^{1}). 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 C_{max} and AUC rules the decision. Even when bioequivalence was not proven for C_{max} and AUC C_{max}/AUC could be within the BE limits. The distribution type of the function is also questionable (ratio of lognormal  paranormal ?). pAUC (partial AUC, truncated AUC), index of early exposure, as was discussed in the parallel thread could be more sensitive than AUC_{last} ('pAUC was always more sensitive than C_{max}'^{2}). The question is what time should be the end time of the calculation? It could be a fixed value (like 72 h), T_{max} or common^{3} T_{last}, T_{low} (AUC_{low}, AUC_{between})... The time should be related to clinically relevant pharmacodynamic measure. What are their advantages? AUMC (first order momentum, the area under the curve 'Concentration*timetime'). If we'll look at the physics analogy  the firstorder momentum for the plain figure is the static momentum which defines the center of mass coordinates, while the secondorder momentum is moment of inertia. The definition could be generalised to the nth 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 approaches^{4} to use MRT instead of T_{1/2} in order to choose the appropriate washout period. C_{apical} 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 1988^{5} and was defined as the arithmetic mean of the concentrations within a 95% confidence interval of C_{max} (i.e., 'not distinguishable from C_{max} at the 5% level of statistical probability by the assay used'). The appropriate time t_{ap lower} and t_{ap 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 C_{max}) and the appropriate time parameters were also mentioned in the article. In the later publications C_{apical} was defined as the arithmetic mean of concentrations at some level (e.g.25% or 50% but not 95% CI!) below C_{max}. 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 C_{apical} 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 (C_{τ}) should serve as an analogue of C_{min} with lack of multiple dose studies. The lag time matters when we deal with delayedrelease formulations. Note that Phoenix defines it as time of observation prior to the first observation with a measurable (nonzero) concentration but not the time to the first observation with a measurable concentration. There could also be other ways^{7} to define it. T_{50%lower}, T_{50%upper}, T_{50%between} (halfvalue duration) or T_{75%lower}, T_{75%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 T_{50%late} or T_{50%early} as intersections. MDT  midpoint duration time (midpoint of halfvalue duration). In the original article^{9} 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: t_{50%lower}+1/2*t_{50%between}, where t _{50%between}  time between the first and the last point where C=50% C_{max} 3). The calculation of MDT could be as follows: t_{50%lower}+1/2*t'_{50%between}, where t'_{50%between}  overall time below C=50% C_{max}. 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?
[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.2019.89.1721 [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 washout 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. Modelindependent estimation of lagtimes with firstorder 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 ModifiedRelease Formulations, The AAPS Journal, Vol. 14, No. 4, December 2012, doi:10.1208/s1224801293968 — "Being in minority, even a minority of one, did not make you mad" 
Helmut ★★★ Vienna, Austria, 20200612 18:23 (1404 d 23:58 ago) @ Astea Posting: # 21534 Views: 6,462 

Hi Nastia, limited time (bloody breadandbutter job). Some desultory thoughts about PK metrics (more to come). ❝ C_{max}/AUC is referred sometimes to be more appropriate in BE studies than C_{max} cause it has much lower variability… Not only that (it’s a sideeffect). What are we interested in? Extent of absoption (cleary AUC) and rate of absorption (k_{a} and possibly t_{lag}). k_{a} is not easily accessible in NCA. C_{max} is a composite surrogate (because influenced by AUC). Easy to show: Define any PK model and vary ƒ whilst keeping k_{a} and t_{lag} constant. C_{max} will change… C_{max}/AUC is an attempt to deal with that. ❝ The distribution type of the function is also questionable (ratio of lognormal  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 lognormal. ❝ pAUC (partial AUC, truncated AUC), index of early exposure, as was discussed in the parallel thread could be more sensitive than AUC_{last} ('pAUC was always more sensitive than C_{max}'^{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 cutoff 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 cutoff 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*timetime'). If we'll look at the physics analogy  the firstorder momentum for the plain figure is the static momentum which defines the center of mass coordinates, while the secondorder momentum is moment of inertia. The definition could be generalised to the nth ❝ 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,\(S_0=\int x^0\cdot f(x)dx = \int f(x)dx\), ❝ MRT is a most common PK parameters that compels to look at the AUMC. There existed approaches^{4} to use MRT instead of T_{1/2} 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 t_{½} 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 AUC_{0–∞} (note that I’m not taking about BE but hardcore PK). For AUMC is should be 99%. I’m quoting Les Benet. Don’t blame me. More to come. — Diftor heh smusma 🖖🏼 Довге життя Україна! _{} Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮 Science Quotes 
Astea ★★ Russia, 20200613 12:17 (1404 d 06:03 ago) @ Helmut Posting: # 21535 Views: 6,343 

Dear Helmut! ❝ Not only that (it’s a sideeffect). What are we interested in? Extent of absoption (cleary AUC) and rate of absorption (k_{a} and possibly t_{lag}). k_{a} is not easily accessible in NCA. C_{max} is a composite surrogate (because influenced by AUC). Easy to show: Define any PK model and vary ƒ whilst keeping k_{a} and t_{lag} constant. C_{max} will change… C_{max}/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 C_{max}/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 C_{max}/AUC (as well as C_{max}) were choosen to be "75,00133,00%") ❝ 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 lognormal. Much easily I should have read something about it somewhere on the forum. Can you please also clarify the distribution of T_{1/2}: 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 T_{1/2} 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 T_{1/2} 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 cutoff 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 'Concentrationtime' (C(t)) are: \(M_t=\int t\cdot C(t)dt\); First is connected with MRT (muliplyed by total square of the curve), and the second should reflect some ideal Concentrationunit parameter (the height of the gravity center). Why not to use it like some alternative to C_{max}? ❝ 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 AUC_{0–∞} (note that I’m not taking about BE but hardcore 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, 20200613 14:28 (1404 d 03:52 ago) @ Astea Posting: # 21536 Views: 6,435 

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 AUC_{0–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 AUC_{0–t(common)}. How early could we stop sampling (x = ?) to get an unbiased estimate is – as you know – on my todo list. I still hold that for IR products it’s much earlier than regulators think. For the rate of absorption? Well, cough. Since C_{max} is a composite metric, it can’t be selective for k_{a}. We don’t have to reinvent the wheel; reading educates.^{1–6} Though I know >50% of the authors personally, I’m not biased. Interesting the abstract^{6} Results: The outcome of a bioequivalence trial was shown to depend on the measure. C_{max}/AUC reflected changes in k_{a}, but not in F. AUC showed dependence on F, but virtually no dependence on k_{a}. For C_{max}, a 3 to 4fold increase in k_{a} and a concomitant 20% decrease in F, as well as corresponding changes in the opposite directions, resulted in bioequivalent outcomes. But thenConclusions: It was concluded that use of C_{max}/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:
Reminds me on another paper^{7} by authors of the FDA assessing the performance of AUC_{0–t} and AUC_{0–∞}. Results: T/Rratios very similar, AUC_{0–∞} more variable. The FDA’s consequences: Use both. ❝ Returning to C_{max}/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 referencescaling chaos. ❝ (In old (e.g. below 2013) russian protocols I even saw the opposite situation when the confidence limits for C_{max}/AUC… I know. Strange. ❝ … (as well as C_{max}) were choosen to be "75,00133,00%") 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 T_{1/2}: 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 t_{½} 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). Additional stuff at the end. ❝ It seemed to me I've read somewhere on the forum that for study planning we should use not the mean T_{1/2} 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 T_{1/2} 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 cutoff 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 'Concentrationtime' (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 Concentrationunit parameter (the height of the gravity center). Why not to use it like some alternative to C_{max}? 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 AUC_{0–∞} (note that I’m not taking about BE but hardcore 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.
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 λ_{z}, 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. — Diftor heh smusma 🖖🏼 Довге життя Україна! _{} Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮 Science Quotes 
Astea ★★ Russia, 20200616 03:33 (1401 d 14:47 ago) @ Helmut Posting: # 21540 Views: 6,035 

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. Riviere^{1}: \(VRT=[\int (tMRT)^2\cdot C(t)dt]/AUC \). It has t^{2} 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 'time^{2}'. 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:
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 hulahoop  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 C^{2} 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=MRT_{ni}MRT_{IV}, where ni is any noninstanteneous administration) and MTT Mean Transit Time^{2}.
[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, 20200616 15:07 (1401 d 03:13 ago) @ Astea Posting: # 21543 Views: 6,031 

Hi Nastia, THX for your response – I have to think about it! 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 lifespan (“mittlere Lebensdauer”) of a total number of N molecules as the arithmetic mean of all times “z_{i}” 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. 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). ❝ ❝ There is a big problem with it. To get a reliable estimate of AUC one has to cover 95% (!) of AUC_{0–∞} (note that I’m not taking about BE but hardcore 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} $$C_p=\frac{A}{k_ak_e}(\textrm{e}^{k_et}\textrm{e}^{k_at}) \tag{6}$$ 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.
— Diftor heh smusma 🖖🏼 Довге життя Україна! _{} Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮 Science Quotes 
mittyri ★★ Russia, 20200617 16:28 (1400 d 01:52 ago) @ Helmut Posting: # 21547 Views: 5,865 

Hi Helmut, ❝ I was wrong (not for the first time).^{ 1} ❝ $$C_p=\frac{A}{k_ak_e}(\textrm{e}^{k_et}\textrm{e}^{k_at}) \tag{6}$$ ❝ 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 MRT_{trunc} 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}*C_{0} <= C_{n}). 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 monoexponential kinetics, the near 2 log period lasts for 6.51 times the elimination halflife<...> By the way I couldn't follow Yamaoka's logic regarding that magic cutoff errors. How did they find it? Regarding prof.Keller's note: he's using IV data for cutoff errors estimation, so it is not clear why he did compare it with 1cpt model with oral firstorder absorption in Yamaoka's article.
— Kind regards, Mittyri 
Astea ★★ Russia, 20200623 16:41 (1394 d 01:39 ago) @ mittyri Posting: # 21565 Views: 5,452 

Dear Helmut! ❝ Reach for the stars, even if you have to stand on a cactus. Susan Longacre 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 t_{½} 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 AUC_{0–∞} (note that I’m not taking about BE but hardcore 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 (1exp(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 t^{2}. If you want to have a constant proportion you need a vessel with a form of parabola x^{4} 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 C_{c} 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 C^{2} in it, it's variability was always lower than that of C_{max}, 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 cutoff errors. How did they find it? I am puzzled with the same question. How did they calculated the time to reach 5% of C_{max}? I slightly modified the Helmut's considerations on the article of Scheerans et al. (2008) Let us consider a onecompartment model with firstorder absorbtion of the form: $$C=\frac{A}{k_ak_e}(\textrm{e}^{k_et}\textrm{e}^{k_at}) \tag{6}$$ then residual area (1AUC_{0t}/AUC_{0inf}) should be as follows: $$AUC_{resid}(x,t)=\frac{x\textrm{e}^{t\cdot k_e}\textrm{e}^{x\cdot k_e*t}}{x1},\quad \textrm{where}\qquad x=\frac{k_a}{k_e}.$$ Let n define the ratio of t to T_{1/2,e}, then $$AUC_{resid}(x,n)=\frac{x\cdot2^{n}2^{nx}}{x1}\sim \frac{x\cdot 2^{n}}{x1}\qquad for\qquad nx>>1. \tag{8}$$ In order to estimate the duration of sampling to achieve specific AUC_{resid} we can use the simplifyed formula $$n=\textrm{log}_2\left(\frac{x}{(x1)AUC_{resid}}\right) \tag{9},$$ for example for x=2 and AUC_{resid}=1% the duration should be n=7.64 T_{1/2}, for AUC_{resid}=20% the duration should be n=3.32 T_{1/2} (the exact value is 3.24.) In particular, $$AUC_{resid}(T_{1/2},x)=\frac{x2^{1x}}{2(x1)};\quad AUC_{resid}(T_{max},x)=\frac{x^{\frac{2x}{1x}}x^{\frac{x}{1x}}}{(x1)};\quad AUC_{resid}(2T_{max},x)=\frac{x^{\frac{3x}{1x}}x^{\frac{2x}{1x}}}{(x1)}. $$ AUC_{resid}(T_{max},x) is a monotone function of x limited from 2/e (0.736) to 1; AUC_{resid}(2T_{max},x) is a monotone function of x limited from 3/e^{3} (0.406) to 1. — "Being in minority, even a minority of one, did not make you mad" 
Helmut ★★★ Vienna, Austria, 20200623 17:55 (1394 d 00:25 ago) @ Astea Posting: # 21566 Views: 5,433 

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.See also there. ❝ ❝ 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 (1exp(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 t^{2}. If you want to have a constant proportion you need a vessel with a form of parabola x^{4} 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 C_{c} 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 C^{2} in it, it's variability was always lower than that of C_{max}, but I should've check it more carefully. Surprises me. Given, I didn’t assess it for ages. Maybe I’m wrong again. — Diftor heh smusma 🖖🏼 Довге життя Україна! _{} Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮 Science Quotes 
Astea ★★ Russia, 20200623 23:41 (1393 d 18:40 ago) @ Helmut Posting: # 21569 Views: 5,330 

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, 20200624 13:29 (1393 d 04:51 ago) @ Astea Posting: # 21572 Views: 5,296 

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 → … 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.rbloggers.com/topologicaltomfooleryinrplottingamobiusstrip library(rgl) — Diftor heh smusma 🖖🏼 Довге життя Україна! _{} Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮 Science Quotes 
Helmut ★★★ Vienna, Austria, 20200613 17:28 (1404 d 00:53 ago) @ Astea Posting: # 21537 Views: 6,327 

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, ❝ C_{apical} 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 1988^{5} […] If C_{apical} 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 (C_{τ}) should serve as an analogue of C_{min} 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 delayedrelease formulations. Note that Phoenix defines it as time of observation prior to the first observation with a measurable (nonzero) concentration but not the time to the first observation with a measurable concentration. There could also be other ways^{7} to define it. Correct. Lagtime 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. ❝ T_{50%between} (halfvalue duration)… AFAIK, the HVD (Half Value Duration) appeared for the first time already in 1974^{1} 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 bible^{2} (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 GBHIworkshops was like a flashback after a bad LSDtrip. 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 Another^{5} 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 Boxenbaum^{6} 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.
— Diftor heh smusma 🖖🏼 Довге життя Україна! _{} Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮 Science Quotes 
mittyri ★★ Russia, 20200615 00:24 (1402 d 17:56 ago) @ Astea Posting: # 21539 Views: 6,169 

Dear Astea, ❝ MRT is a most common PK parameters that compels to look at the AUMC. There existed approaches^{4} to use MRT instead of T_{1/2} in order to choose the appropriate washout period. Just my 2 cents regarding MRT. You cannot estimate it for nonIV ways of administration. For nonIV that parameter is called 'gravity duration', since includes both MIT and MRT. See here for discussion. — Kind regards, Mittyri 
ElMaestro ★★★ Denmark, 20200616 11:46 (1401 d 06:34 ago) @ Astea Posting: # 21541 Views: 6,002 

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 lognormal  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 lognormals is normal (or lognormal, 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, 20200616 12:54 (1401 d 05:26 ago) @ ElMaestro Posting: # 21542 Views: 6,021 

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, 20200616 15:14 (1401 d 03:06 ago) @ mittyri Posting: # 21544 Views: 5,956 

— Diftor heh smusma 🖖🏼 Довге життя Україна! _{} Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮 Science Quotes 
ElMaestro ★★★ Denmark, 20200620 12:33 (1397 d 05:47 ago) @ mittyri Posting: # 21551 Views: 5,653 

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, 20200621 01:04 (1396 d 17:16 ago) @ ElMaestro Posting: # 21553 Views: 5,605 

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 — Kind regards, Mittyri 