PharmPK Discussion - Two compartment infusion equation

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• On 17 Jul 2006 at 00:28:39, "Peter CMAlt" (petercmalt.-at-.hotmail.com) sent the message
`Dear friendsI need some help regarding the solution of the 2 compartment openmodel with iv infusion input. I cannot find the way to pass from theform =[(K0/s)(s+K21)]/[(s+a)(s+b)] to the time equation, also thereseems to be some different solutions. Can anyone help me with thesolution or with some books where I can go and find the solutionThanks to allBest regardsPeterCmalt[Have you tried the fingerprint method - see http://www.boomer.org/c/p3/c07/c0702.html and references listed - db]`
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• On 17 Jul 2006 at 14:33:57, "Porzio, Stefano" (Stefano.Porzio.aaa.ZambonGroup.com) sent the message
`The following message was posted to: PharmPKDear PeterCmalt,if I correctly understand your problem, you need  the solution forthe two-open model with central compartment elimination (I suppose)and zero-order input (infusion is a zero order input).If you need peripheral elimination, obviously the solution is different.You can find the solution for central compartment elimination and, ifrequired a lot solutions for many other compartmental models, inPharmacokinetics for the Pharmaceutical ScientistsJohn G. Wagner, PhDTechnomic Publishing Company Inc. (1993)In the case of the two-open model with central compartmentelimination (I suppose) and zero-order input, I tray to report theWagner solution (it's difficult to write equation in plain test....):"During the zero-order input, such as intravenous infusion, theconcentration C in the central compartment at the time t, during theinterval 0 < t < T where T is the duration of the zero-order input,is given by the equationC= Ko (K21-L1)(1-exp(-L1t))/L1(L2-L1)Vc + K0(K21-L2)(1-exp(L2t))/L2(L1-L2)VcAfter the zero-order input ceases (i.e. when t >T), the concentrationis given by the equationC= Ko (K21-L1)(exp(L1T)-1)exp(-L1t)/L1(L2-L1)Vc + Ko (k21-L2) (exp(L2T)-1)exp(-L1t)/L2(L1-L2)VcIn the previous two equationsKo ist the zero-order input constant,K21 is the transfer rate constant from peripheral compartment to thecentral compartmentL1 is  the first elimination constant ( the first macro-constant ofthe bi-exponential disposition)L2 is  the second elimination constant ( the second macro-constant ofthe bi-exponential disposition)Vc is the volume of the central compartment"I hope this is useful!Best regardsStefano`
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• On 17 Jul 2006 at 12:10:40, "MONZANI VALMEN" (V.MONZANI.at.italfarmaco.com) sent the message
`The following message was posted to: PharmPKDear PeterThe book I found to treat the argument deeply is Pharmacokineticssecond edition by M.Gibaldi and D.Perrier.Hope this helpValmen`
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• On 21 Jul 2006 at 12:26:39, "Peter CMAlt" (petercmalt.at.hotmail.com) sent the message
`Dear FriendsThanks a lot for your answers, however I should say that my questioncontinues. I probably should be more specific (let's see if i canwrite the problem with the equations in plain text).The solution of the two compartmet open model with iv infusion inputwill provide the following equation:mass1=[(K0/s)(s+K21)]/[(s+a)(s+b)] .Using the fingertip method as sugested by David Bourne (or usingdirect Laplace transformations) to pass for the time order gives theequation:(1)  M(t)=(K0K21/ab)+[(K0(K21-a)/a(a-b))e(-at)]+[(K0(K21-b)/b(b-a))e(-bt)] (i hope i didn't miss anything)The final equation provided by Wagner and other authors is(2) M(t)= K0(K21-a)(1-exp(-at))/a(b-a) + K0(K21-b)(1-exp(-bt))/b(a-b)So my problem starts in the passagem from (1) to (2). I think this isonly a problem of mathemathical simplication, or lack of personalmathemathical knowledge, but can anyone has the kind to help me (iknow that this is too much but with much detail as possible!!!)?Thak you very much for your helpBest regardsPeterCmalt`
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• On 15 Aug 2006 at 19:10:46, "Peter CMAlt" (petercmalt.-at-.hotmail.com) sent the message
`Dear FriendsRegarding the solution of the two compartment open model withperfusion there still are some questions:After applying the fingertip method to the equation:[(K0(s+K21)]/[s(s+a)(s+b)]I obtain the equation:(K0K21/ab)+{[K0(K21-a)]/[a(a-b)]}exp(-at)-{[K0(K21-b)]/[b(a-b)]}exp(-bt)My problem is that I cannot pass from the previous equation to theone presented in the Wagner book:{[K0(K21-a)]/[a(a-b)]}(1-exp(-at))+{[K0(K21-b)]/[b(a-b)]}(1-exp(-bt))Can anyone help me?Thanks in advance for your kind helpPeterCmalt`
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• On 17 Aug 2006 at 10:43:40, Rakesh.2.Nagilla.at.gsk.com sent the message
`Peter,The actual equation from Wagner's book isk0(k21-a)                 k0(k21-b)--------- x [1-exp(-at] + --------- x [1-exp(-bt)]a(b-a)                    b(a-b)[retyped from submitted figure - db]which when simplified would lead to the equation you obtained.`
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• On 17 Aug 2006 at 15:46:27, "Shawn D. Spencer" (shawn.spencer.aaa.famu.edu) sent the message
`Peter,It is apparent to me, that J. Wagner used the Laplace tables inarriving at his function,  whereas heaviside expansion and partialfraction theorems would arrive at a different, albeit it equivalent,function.  Not every function with two or more exponential terms canbe manipulated algebraically, as some are transcendental.  I do notknow if this is the case here, however if you can simulate bothequations, then you have a suitable endpoint to move forward.Regards,Shawn D. Spencer, Ph.D., R.Ph.Assistant Professor of Biopharmaceutics and PharmacokineticsCollege of Pharmacy and Pharmaceutical SciencesFlorida A&M UniversityTallahassee, FL 32307shawn.spencer.-at-.famu.edu `
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