# PharmPK Discussion - Tmax oral two compartment model

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• On 25 Mar 2011 at 23:17:00, DHAIVAT PARIKH (dhaivatpharma.at.gmail.com) sent the message
`I am in search of equations alongwith its derivation for Value of Tmax  (also Cmax ifpossible) for Two Compartment Open Pharmacokinetic Model  for Oral Administration, whichis not available in majority of standard  text books of Biopharm & PharmacokineticsOne equation i obtain from internet is Tmax = (Ln (a/b))/(a-b) , which  may be true but iam not sure.where, a and b (also described as Alpha and Beta as well as Lamda1 and  Lamda2 in otherliteratures) are Distribution Rate Constant &  Elimination Rate Constant respectivelyb is also obtained from Elimination (terminal) half lifeKindly inform me that whether the above equation for Tmax is true or  not.if not then what is right equationand if True then please guide me that from where i will find the  derivation of entireequation.Anticipating your kind co-operation and prompt response.Thanks & Regard`
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• On 25 Mar 2011 at 19:50:16, "Lee, Jee Eun" (JeeEun.Lee.aaa.fda.hhs.gov) sent the message
`The following message was posted to: PharmPKHi,Tmax is defined as the time to reach Cmax, the maximal PLASMA  concentration. Forone-compartment first-order absorption model, Tmax  can be written as ln(ka/ke)/(ka-ke)where ka is the first-order  absorption rate constant and ke is the first-orderelimination rate  constant. The equation for plasma concentration at time t is describedbelow and Tmax was derived by finding t when dCp/dt =0.Cp(t)=F*(Dose/Vd)*(ka/(ka-ke))*{exp(-ke*t)-exp(-ka*t)}dCp/dt=F*(Dose/Vd)*(ka/(ka-ke))*{-ke*exp(-ke*t)+ka*exp(-ka*t)}(Try to see if you get ln(ka/ke)/(ka-ke), then you will understand  easily for the Tmax oftwo-compartment first-order absorption model)If Tmax is written as ln(alpha/beta)/(alpha-beta), then it implies the  model of interestis two-compartment intravenous bolus model. Alpha and  beta are hybrid terms for thosethree micro constants of the  two-compartment model (k10, k12, k21) and the hybrid termswere created  for the purpose of solving Laplace transformation. I do not see itspractical value in this case because the Tmax has been derived for the  time to reachmaximal PERIPHERAL concentration following an intravenous  bolus dose. Would you need it?What you need is the time to reach the maximal PLASMA concentration for  two-compartmentfirst-order absorption model, then the Tmax will be time  t when dCp/dt=0 from theequation below. dCp/dt=F*(Dose/Vc)*ka{-ka*((k21-ka)/(alpha-ka)*(beta-ka))*exp(-ka*t)-alp ha*((k21-alpha)/(ka-alpha)*(beta-alpha))*exp(-alpha*t)-beta*((k21-beta)/(k a-beta)*(alpha-beta))*exp(-beta*t)}where ka is the first-order absorption rate constant, k21 is the  first-order transferrate constant from peripheral compartment to  central compartment, Vc is central volume ofdistribution, alpha and  beta are hybrid terms for k12 (the first-order transfer rateconstant  from central compartment to peripheral compartment), k21, and k10 (thefirst-order elimination rate constant from central compartment).It seems difficult to simplify this equation like the case of  one-compartment model, thusI do not see its practical utility unless  you want to compare some extreme cases.Cmax can be obtained by simple plug-in of the Tmax obtained above into  the equation belowfor the plasma concentration at time t.Cp(t)=  F*(Dose/Vc)*ka{((k21-ka)/(alpha-ka)*(beta-ka))*exp(-ka*t)+((k21-alpha)/(ka -alpha)*(beta-alpha))*exp(-alpha*t)+((k21-beta)/(ka-beta)*(alpha-beta))*ex p(-beta*t)}I hope this helps even though it does not give you a simple solution.I attempted to derive these equations for the first time, so any  comments for my possiblemistakes will be greatly appreciated.Thanks,Jee EunDisclaimer: This is my personal opinion and does not represent my  position at FDA.`
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• On 25 Mar 2011 at 17:09:07, "Benet, Leslie" (Leslie.Benet.aaa.ucsf.edu) sent the message
`The following message was posted to: PharmPKThe equation you list is for a one compartment model where a and b are  the absorption andelimination rate constants.Tmax can not be solved by an equation following absorption in a two  compartment model.It can only be determined by simulation with  explicit PK parameters.Leslie Z. Benet, Ph.D.ProfessorDepartment of Bioengineering & Therapeutic Sciences Schools of Pharmacy & MedicineUniversity of California San Francisco533 Parnassus Avenue, Room U-68San Francisco, CA 94143-0912`
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• On 25 Mar 2011 at 18:24:47, Roger Jelliffe (jelliffe.-at-.usc.edu) sent the message
`The following message was posted to: PharmPKDear All:  About this model with Ka, V, and Ke, it is interesting to see thatthe tmax also depends on the amount or concentration (however you wish towrite it) of drug present at the time the dose is given. For example, if youwish to compute a dosage regimen to hit a desired peak during each ofseveral dosing intervals, when the initial condition is zero, as with thefirst dose, the Tmax is latest, and the trough after that first loading doseis the highest. Because of this, the second dose is always the smallest, andthe peak is the earliest. After that, the doses get a little bigger andthings begin to settle down and stabilize.  So the initial condition at thetime of the dose must also be taken into account, I forget just how.Very best regards,Roger Jelliffe`
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• On 26 Mar 2011 at 16:08:34, Helmut Schuetz (helmut.schuetz.-a-.bebac.at) sent the message
`The following message was posted to: PharmPKDear Dhaivat!> I am in search of equations alongwith its derivation for Value of Tmax> (also Cmax if possible) for Two Compartment Open Pharmacokinetic Model> for Oral Administration, which is not available in majority of standard> text books of Biopharm&  Pharmacokinetics>Right. Not missing in the majority of textbooks, but in all of them (inother words, stop searching).> One equation i obtain from internet is Tmax = (Ln (a/b))/(a-b) , which> may be true but i am not sure.>Correct only for a one-compartment model (a = absorption rate constant,b = elimination rate constant).> if not then what is right equation> and if True then please guide me that from where i will find the> derivation of entire equation.>You are already at the right site. Derivation of the one-compartment here:http://www.boomer.org/c/p4/c08/c0802.html and followingsNow for the bad news. Cmax/Tmax is the point of the curve where thefirst derivate (i.e. the slope) is zero. In order to calculate Tmax, youhave to get the first derivate of C(t) and find the root of thisfunction. Before Tmax dC/dt >0, at Tmax dC/dt = 0, and after Tmax dC/dt<0. For more than two exponential terms (i.e. > one-compartments) thederivative of C(t) cannot be solved in closed form. That's why I wroteabove "stop searching". You can only estimate Cmax/Tmax by an iterative(numeric) algorithm.Helmut-Ing. Helmut SchuetzBEBAC - Consultancy Services forBioequivalence and Bioavailability StudiesNeubaugasse 36/111070 Vienna, Austriae-mail  helmut.schuetz.-a-.bebac.atweb     http://bebac.atcontact http://bebac.at/Contact.htmforum   http://forum.bebac.at`
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• On 26 Mar 2011 at 11:48:39, "Pereira, Luis" (Luis.Pereira.-at-.childrens.harvard.edu) sent the message
`The following message was posted to: PharmPKDear Dhaivat and AllThe analytical derivation for tmax comes from equating the first  derivative of the C(t)expression to zero and solving for time, as done  to estimate the extrema (minima ormaxima) of any function. The  distinction depends on the sign of the second derivative,but since in  PK often C(t) profiles are biphasic, thus the generalization about tmax  andCmax. However, not all expressions, i.e. compartmental models, have  an analyticalsolution. This was extensively discussed by JG Wagner when  he coined the term 'vanishingexponentials' (JPB (1976) 4:5,395; I  recommend). Basically, for a single dose, the timeto reach the highest  concentration depends on the overall rate limiting step, i.e. for atwo  compartment disposition, the relative magnitude of ka (assuming first  orderabsorption), alpha and beta. They are often not all identifiable  with real data, and thustmax does not have a single universal  expression, unlike with the single exponentialdisposition case.  However, even in that case, tmax will only be larger than zero if ka>ke which is an assumption, not a fact (careful with flip-flop, controlled  release, etc.).For multiple doses, depending on the accumulation  factors (AF) for both exponentials, inthis case, tmax will be different  for each dose, and at steady-state instead ofln(ka/ke)/(ka-ke) it  becomes ln(ka.AFabs/ke.AFel)/(ka-ke) which is always a smallernumber.  The expression ln(alpha/beta)/(alpha-beta) does correspond to a tmax but  for C2,the estimated concentration in the peripheral compartment, again  assuming first orderkinetics.Hope this helps,Cheers--Luis Pereira, PhDAnesthesia DepartmentChildren's Hospital BostonHarvard Medical School[Wagner. Linear pharmacokinetic models and vanishing exponential terms:  implications inpharmacokinetics. J Pharmacokinet Biopharm (1976) vol. 4  (5) pp. 395-425 - db]`
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