# PharmPK Discussion - Emax and Cmax - parameters and statistics

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• On 19 Jun 2012 at 09:30:24, Nick Holford (n.holford.-a-.auckland.ac.nz) sent the message
`Walt,You wrote:> I'm confused by your statement regarding Emax and Cmax. Both are potentially> physically observable quantities (if we could time the measurements> perfectly). Unlike derived pharmacokinetic variables like volume of> distribution or peripheral compartment rate constants, concentration is as> measurable as effect, is it not?My reply:Cmax is an observable quantity which is the reason why it is a poordescriptor of the true maximum concentration. I call this apharmacokinetic statistic. Like Tmax and AUC (its SHAManalysis%29-%28SHAM%29.html>efulbrothers) these are descriptive statistics which may be used to estimateparameters. Note that the average is a descriptive statistic which is anestimate of a parameter (mean) of a distribution. AUC on the other handis not an estimate of a pharmacokinetic parameter because it requiresanother variable (dose) and another parameter (bioavailability toestimate clearance or clearance to estimate bioavailability) to beuseful for estimating a parameter.Emax is not an observable quantity. It is a parameter of the Emaxpharmacodynamic model. Because the effect is only equal to Emax atinfinite concentration (according to the theory of the model) then it isclear that Emax can never be observed in any practical sense. I teachthis whenever I introduce the concepts of pharmacodynamics (see slides 6and 7 inhttp://holford.fmhs.auckland.ac.nz/docs/immediate-time-course-of-drug-effect.pdf).On the other hand it is possible (by very good luck) to sample andmeasure a concentration at the precise time of the true maximumconcentration. In that special case the measured Cmax would be anestimate of the true Cmax - the parameter describing the concentrationprofile. As David points out the measured Cmax could be either higher orlower than the true Cmax due to measurement error.What you call 'derived pharmacokinetic variables' I prefer to refer toas parameters. Please see Alman & Bland (1999) for a readableexplanation of the difference between variables and parameters.1.    Altman DG, Bland JM. Statistics notes: variables and parameters.BMJ 1999; 318: 1667.`
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• On 19 Jun 2012 at 11:55:15, Walt (walt.-a-.simulations-plus.com) sent the message
`The following message was posted to: PharmPKDimiter and Prasad,Yes, universities teach much more than what is needed for a particular job.I think the best ones emphasize critical thinking skills over rotememorization of formulae.That's one thing we like to emphasize in our advanced (and even basic)training course - what to do first before running simulation and modelingsoftware - stand back and look at the data! Think about what it's trying totell you before you start plugging numbers into software and generatingplots and tables. That's where the emphasis on Cmax always being higher thanthe highest observed data point comes in. So many times we fit a model to aset of data and the model Cmax is between two data points and higher -either a little or a lot. Some students (these are largely Ph.D. industryscientists!) think the model is wrong if the peak of the curve is above thehighest observation.I love to set up a trap and show some data and ask "What is Cmax". Worksevery time! Then remove that data point and ask, "OK, now what is Cmax?" andwatch the look on their faces. Then fit models with and without that datapoint and see they're nearly the same, and the predicted Cmax values arevery close wither way. They go away remembering that Cmax is higher than thehighest observation so often that you might as well bank on it.Of course, we also teach that once a model has been fitted and looks pretty,challenge yourself to ask what could be making it look nice, but still beincorrect. One of my favorite sayings is "It's not about making the blueline go through the little squares" (the blue line is the simulation and thelittle squares are the observations on a GastroPlus Cp-time plot). It'sabout understanding, to the extent possible with the available data,mechanistically what is going on in the system.Best regards,WaltWalt WoltoszChairman & CEOSimulations Plus, Inc. (NASDAQ: SLP)42505 10th Street WestLancaster, CA  93534-7059U.S.A.http://www.simulations-plus.com`
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• On 19 Jun 2012 at 14:07:11, Walt (walt.-a-.simulations-plus.com) sent the message
`The following message was posted to: PharmPKHelmut,You said:"I disagree. Due to random variability the observed Cmax may be higher thanthe theoretical one (acceptable inaccuracy of the bioanalytical method inthe upper range 15% with imprecision of 15%). I have used many modelingpackages in the last 30+ years. The absorption phase is difficult to modeleven if enough data points are available and your credo "a single ka doesnot exist" is built in the model. In many (most?) instances the estimatedCmax is lower than the observed one - which is counterintuitive. I'm on yourside following the theoretical argument that we likely miss the 'true' Cmax(sampling too early or too late) and the observed one should be lower thanthe fitted one. Unfortunately my experiences tell another story. Too badthat software sometimes don't works according to our expectations."I was not referring to the theoretical (or estimated) Cmax, which can ofcourse be lower than the maximum observation, depending on the fittingmethod and weighting function (unweighted tends to result in higherpredicted Cmax than weighting using the usual schemes that are not based onvariance). I was referring to what is actually the maximum concentration inthe system. No matter how it is measured, the variances apply to themeasuring methodology, and it would be exceptionally fortuitous to take ameasurement at the precise instant of maximum concentration.Best regards,WaltWalt WoltoszChairman & CEOSimulations Plus, Inc. (NASDAQ: SLP)42505 10th Street WestLancaster, CA  93534-7059U.S.A.http://www.simulations-plus.com`
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• On 20 Jun 2012 at 21:48:29, Dimiter Terziivanov (terziiv.aaa.yahoo.com) sent the message
`Dear Walt,I adhere to your note. Variables/paramaters with discrete distributions, as Tmax,are difficult to understanding by students. I have no evidence-based explanationbut rather a naive one arising from our every day experience with continuousvariables. As your well-designed example justifies, application of simulationapproaches in student education might be of help in overcoming the habit to dealmore frequently with continuous variables.Regards,DimiterDimiter Terziivanov, MD,PhD,DSc, ProfessorDept. of Pharmacology and Clinical PharmacologySOFIA UNIVERSITY"ST. KLIMENT OHRIDSKI"FACULTY OF MEDICINEUNIV HOSP "LOZENETZ"1 Koziak str.1407 Sofia, BULGARIA`
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• On 22 Jun 2012 at 09:28:08, Walt (walt.-at-.simulations-plus.com) sent the message
`The following message was posted to: PharmPKDimiter,Thanks for your comment. I always thought parameters were numbers in anequation that were typically fitted to make the equation match something inthe real world that was a variable. For example, in the equation for astraight line:y = mx + bI would have thought y and x were variables, and m and b were parameters.But if I understood Nick's comment correctly, it seemed that they werereversed.Best regards,WaltWalt WoltoszChairman & CEOSimulations Plus, Inc. (NASDAQ: SLP)42505 10th Street WestLancaster, CA  93534-7059U.S.A.http://www.simulations-plus.com[I think Cmax and Emax are parameters, E and C are the variables. - db]`
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• On 22 Jun 2012 at 23:23:54, Dimiter Terziivanov (terziiv.-at-.yahoo.com) sent the message
`Walt and David,I share your understanding regarding variables and parameters. In real life wedeal with variables, not with parameters. When starting a scientific study ofset of variables we aim at their parameterization in order to establish somequantitative relationships. In this context, according to my understanding, 'y'and 'x' and 'Concs' and 'Effect' are measurable variables, whereas 'm' and 'b'and 'Cmax' and 'Emax' are computed parameters.Kind regards,DimiterDimiter Terziivanov, MD,PhD,DSc, ProfessorDept. of Pharmacology and Clinical PharmacologySOFIA UNIVERSITY"ST. KLIMENT OHRIDSKI"FACULTY OF MEDICINEUNIV HOSP "LOZENETZ"1 Koziak str.1407 Sofia, BULGARIA`
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• On 25 Jun 2012 at 22:30:36, Walt (walt.at.simulations-plus.com) sent the message
`The following message was posted to: PharmPKDimiter,I guess in my mind I still think of Cmax and Emax as the elusive _actual_maximum concentration and effect, although neither may ever be measured.What we tend to call Cmax and Emax are estimates, but I don't see estimatesas the same as parameters, which I think of as coefficient, exponents, andthe like in empirical equations that are used to try to model things likeconcentrations and effects. Perhaps we should do something like using the Cprefix to refer to calculated values as is done with ClogP, e.g., CCmax,CEmax (ugly, eh?!) or some other way to distinguish between the _true_ Cmaxand Emax and our best estimates of what they are.Best regards,Walt`
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• On 26 Jun 2012 at 18:48:33, Nick Holford (n.holford.aaa.auckland.ac.nz) sent the message
`Walt,The statisticians have 'been there done that' a long time ago.The usual convention is put a hat (^) on an estimate or to use a greek alphabetfor the true value and a latin alphabet for an estimate.While it is possible to parameterize a PK model in terms of Cmax its not reallyuseful and the algebra is tricky.  So Cmax will nearly always be a statistic(the empirically highest measured value) which you refer to as a 'calculatedvalue'. This is neither an estimate nor a parameter. Emax on the other handcannot be described as a statistic. It is only possible to imagine it as aparameter which may have an estimate derived from the data.Nick`
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• On 27 Jun 2012 at 21:59:05, Walt (walt.-at-.simulations-plus.com) sent the message
`The following message was posted to: PharmPKNick,Thanks for your response. But I'm afraid there continues to be amiscommunication here.I would not parameterize a model in terms of Cmax, and I would never assignthe highest measured value to Cmax, which was my original point - the trueCmax is always somewhat higher than the highest measured value. Because themeasurement taken closest to the actual Cmax is always either before orafter the actual Cmax, sometimes by a small amount, sometimes by a largeamount. In general, I believe we can safely say that the true maximumconcentration is greater than any measurement. Because a well-fitted modelwill often predict a higher maximum concentration than the highest observedconcentration, we can estimate Cmax better with a well-fitted model than byusing the maximum observation. I do not refer to Cmax as a calculated value.I refer to Cmax as the elusive actual maximum value.As far as Emax, I believe there should typically be a maximum effect, whichmight be a result of the maximum absorbable dose, saturation of the targetreceptor, adverse effect terminally overcoming therapeutic effect, or someother factor. Maybe it can't be measured easily, if at all, but it seemsthere should be some maximum therapeutic effect achievable with any drug.Estimating it is probably almost always, as you say, a statistical exercise,but nonetheless, there should be some maximum effect for any drug.Best regards,Walt`
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• On 27 Jun 2012 at 21:59:05, Walt (walt.-a-.simulations-plus.com) sent the message
`The following message was posted to: PharmPKNick,Thanks for your response. But I'm afraid there continues to be amiscommunication here.I would not parameterize a model in terms of Cmax, and I would never assignthe highest measured value to Cmax, which was my original point - the trueCmax is always somewhat higher than the highest measured value. Because themeasurement taken closest to the actual Cmax is always either before orafter the actual Cmax, sometimes by a small amount, sometimes by a largeamount. In general, I believe we can safely say that the true maximumconcentration is greater than any measurement. Because a well-fitted modelwill often predict a higher maximum concentration than the highest observedconcentration, we can estimate Cmax better with a well-fitted model than byusing the maximum observation. I do not refer to Cmax as a calculated value.I refer to Cmax as the elusive actual maximum value.As far as Emax, I believe there should typically be a maximum effect, whichmight be a result of the maximum absorbable dose, saturation of the targetreceptor, adverse effect terminally overcoming therapeutic effect, or someother factor. Maybe it can't be measured easily, if at all, but it seemsthere should be some maximum therapeutic effect achievable with any drug.Estimating it is probably almost always, as you say, a statistical exercise,but nonetheless, there should be some maximum effect for any drug.Best regards,Walt`
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• On 7 Jul 2012 at 09:58:41, Nick Holford (n.holford.aaa.auckland.ac.nz) sent the message
`Walt,On 28/06/2012 5:59 a.m., Walt wrote:> I would not parameterize a model in terms of Cmax, and I would never assign> the highest measured value to Cmax,> As far as Emax, I believe there should typically be a maximum effect, which> might be a result of the maximum absorbable dose, saturation of the target> receptor, adverse effect terminally overcoming therapeutic effect, or some> other factor. Maybe it can't be measured easily, if at all, but it seems> there should be some maximum therapeutic effect achievable with any drug.I agree with you about parameterizing a model in terms of Cmax. Itstechnically possible but of course is dependent on the dose so it is notreally a parameter except in the limited case of a fixed dose.On the other hand the most widely used definition of Cmax (e.g. forregulatory bioequivalence) is the highest measured value. I am quitehappy with this definition as long as you don't try to call it aparameter. It is a statistic -- just like its SHAM brothers Tmax and AUC.With regard to the Emax parameter, I want to make a clear distinctionbetween the highest actual effect of a drug (which I like to call thepeak effect) and the asymptotic value predicted at infiniteconcentration by the Emax model (this unattainable value is Emax). Emaxis a model parameter which can be estimated but never observed (bydefinition). In your example of an effect being limited by the 'maximumabsorbable dose' then the highest effect is the peak effect but not Emax.On the other hand effects which are limited by receptor saturation willasymptotically reach Emax. Receptor saturation is of course anotherexample of a non-observable phenomenon because it can only occur atinfinite ligand concentration.Best wishes,Nick`
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• On 8 Jul 2012 at 14:39:27, Walt (walt.-at-.simulations-plus.com) sent the message
`The following message was posted to: PharmPKNick,Thanks for your reply, in which you said,"I agree with you about parameterizing a model in terms of Cmax. Itstechnically possible but of course is dependent on the dose so it is notreally a parameter except in the limited case of a fixed dose."On the other hand the most widely used definition of Cmax (e.g. forregulatory bioequivalence) is the highest measured value. I am quite happywith this definition as long as you don't try to call it a parameter. It isa statistic -- just like its SHAM brothers Tmax and AUC."Perhaps it's a matter of semantics to some extent. To an engineer, a valuethat is purported to represented the maximum or minimum of a quantity (witha subscript of max or min) is customarily assumed to represent the actualmaximum or minimum, regardless of what may have been measured. When I workedon the Space Shuttle program, for example, Qmax was the (real) maximumdynamic pressure, Vmax was the (real) maximum velocity, (L/D)max was themaximum lift-to-drag ratio, and so on. Of course, we had a much betterability to measure these values because our sampling rates were virtuallycontinuous.So Cmax and Emax seemed to me to represent the actual maximum values withinthe system, regardless of whether or not they could ever be measured. We canestimate them using various methods, but for these particular values, it istheoretically impossible to measure either one - Emax for the reason youmentioned and Cmax because it would take an extremely fortuitous measurementat exactly the right time to measure it.So unless there is some other notation for the actual maximum value, I thinkit's misleading to say Cmax is the highest observed value, because we knowthat in very many (if not the majority) of cases, the highest measured valuewill be below Cmax by a significant amount. Certainly the highest observedvalue will not be greater than the actual maximum (ignoring measurementerror for the moment). A _properly fitted_ model will provide a betterestimate than the highest observed value. It's quite easy to take a plasmaconcentration-time data set, fit a PK model, and then remove the highestobserved value and refit the model and get nearly the same peak for thefitted model, as we show in our training courses. Even deleting the twohighest values results in a far better estimate than the highest (remaining)observation.So why would anyone want to trust that the highest observed value is alwaysa good estimate of the real Cmax?Best regards,WaltWalt WoltoszChairman & CEOSimulations Plus, Inc. (NASDAQ: SLP)42505 10th Street WestLancaster, CA  93534-7059U.S.A.http://www.simulations-plus.com`
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