# PharmPK Discussion - Laplace transform derivation of infusion equation

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• On 11 Sep 2009 at 19:41:48, David Bourne (david.-at-.boomer.org) sent the message
`The following message was posted to: PharmPKHello allI'm stuck... I set a exercise to derive the equation for X(or Cp)versus time using a general infusion equation and ended up not beingable to complete the Laplace based derivation myself ;-( From http://www.boomer.org/c/p4/c02/c07/c0707.htmlor Benet's paper reference in the footerIV infusion, drug amount in the central compartment, one compartmentgives the Laplace                 -a x s    -z x s          k0 x (e       - e      )       1X(bar) = ------------------------ x ---------                   s                (s + kel)Roots = 0, and -kelFinger print method seems to give      k0 [ z x kel    ]    -kel x tX = --- [e        - 1] x e     kel [            ]which doesn't seem to be the same as the expected      k0       -kel x z     -kel x tX = --- [1 - e        ] x e     kelHelp!where did I do wrong? :-)Thanks, David`
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• On 11 Sep 2009 at 23:12:14, Leonid Gibiansky (LGibiansky.at.quantpharm.com) sent the message
`The following message was posted to: PharmPKDavid,Your derivation is correct (assuming a=0) but your expectation iswrong :)To see is, set t=z when z is very large. You should get k/kel (asfollows from the steady-state equation and from your Laplacederivation) but your expectation would give incorrect value (zero).Leonid--Leonid Gibiansky, Ph.D.President, QuantPharm LLCweb:    www.quantpharm.come-mail: LGibiansky at quantpharm.com--Thanks LeonidI did assume a = 0So my derivation >      k0 [ kel x z    ]    -kel x t > X = --- [e        - 1] x e >     kel [            ]is correct. If z = t (continuous infusion) this transforms into      k0 [ kel x t    ]    -kel x tX = --- [e        - 1] x e     kel [            ]         k0 [ (t x kel - t x kel)    -kel x t]or X = --- [e                    - e        ]        kel [                                ]         k0 [     -kel x t]or X = --- [1 - e        ]        kel [             ]And when t > z my derivation is another form of the equation for Xafter the infusion has stopped.      k0 [     -kel*z]    -kel x (t-z)X = --- [1 - e      ] x e     kel [           ](Checked numerically :)Great, thanks again.`
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• On 14 Sep 2009 at 15:06:02, (maria.durisova.-a-.savba.sk) sent the message
`The following message was posted to: PharmPKHello,you can use the method described here:http://www.uef.sav.sk/CXT-Mainto complete the Laplace based derivation.Regards,Maria Durisova`
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