# PharmPK Discussion - Cmax and Cmin

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• On 1 Oct 2004 at 10:44:10, "Daniel Rossi de Campos" (Daniel.Campos.-at-.anvisa.gov.br) sent the message
`Dear All,Could anyone provide me equations to find values of Cssmax and Cssminafter a oral dose. There is any free program that could do it.Regards,Daniel[After one dose Cpmin = 0 ;-) Cpmax at tmax (=tpeak) fromhttp://www.boomer.org/c/p4/c08/c0803.htmlAt steady state Cpmin can be calculated fromhttp://www.boomer.org/c/p4/c15/c1503.html with various assumptions.Cpmax can be estimated unless you integrate the multiple dose equation(I haven't yet ...) or solve numerically searching for the tpeak atsteady state - db]`
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• On 4 Oct 2004 at 09:33:21, "Hans Proost" (j.h.proost.-at-.farm.rug.nl) sent the message
`Dear Daniel,You wrote: > Could anyone provide me equations to find values of Cssmax and Cssmin > after a oral dose. There is any free program that could do it.The equation for the concentration at steady state can be deriveddirectlyfrom the single dose equation:C(t) = F * Dose / V * ka / (ka - k) * (exp(-k*t) - exp(-ka*t))results in:Css(t) = F * Dose / V * ka / (ka - k) * (exp(-k*t) / (1- exp(-k*tau)) -exp(-ka*t) / (1 - exp(-ka*tau)))at steady state with doses Dose at dosing interval tau.Cminss is obtained by substituting t = 0 (or t = tau; this should givethesame result).The maximum concentration is reached if dC/dt = 0, thus:dC/dt = F * Dose / V * ka / (ka - k) * (-k * exp(-k*t) / (1-exp(-k*tau)) +ka * exp(-ka*t) / (1 - exp(-ka*tau))) From dC/dt = 0 it follows after replacing t with Tmax:-k * exp(-k*Tmax) / (1- exp(-k*tau)) + ka * exp(-ka*Tmax) / (1 -exp(-ka*tau)) = 0which reduces to:Tmax = ln(ka / k * (1- exp(-k*tau)) / (1 - exp(-ka*tau))) / (ka - k)Substituting Tmax in Css(t) results in Cmaxss.Please note:- Tmax at steady state is different from (i.e. shorter than) Tmax afterasingle dose.- these equation work for both ka > k and ka < k (not for ka = k).Best regards,Hans ProostJohannes H. ProostDept. of Pharmacokinetics and Drug DeliveryUniversity Centre for PharmacyAntonius Deusinglaan 19713 AV Groningen, The Netherlandstel. 31-50 363 3292fax  31-50 363 3247Email: j.h.proost.at.farm.rug.nl`
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