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The following message was posted to: PharmPK
Group,
Can someone post an equation to express the area under effect vs. time
curve (AUEC or AUE) in the term of area under plasma concentration
curve (AUC)?
Assume the PD effect follows a simple Emax/EC50 model:
E(t) = Emax*C(t)/(EC50 + C(t)), where C(t) is the PK curve.
Thanks,
Jack
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Dear Jack,
no one can post such an equation, since in general (i.e. without very
specific assumptions on C(t)) it does not exist.
It cannot exist, because AUE is not a unique function of AUC.
To demonstrate this, assume a very simple case:
Emax=1
EC50=1
and consider two concentration-time profiles, such that C2(t)=0.5*C1(t/
2) for any t. The only additional assumption on C1 is the AUC1 exist
at all, besides that the profile may be arbitrary.
Using basic principles of integral calculus you find that AUC1=AUC2,
but AUE2 > AUE1 (for a detailed proof see below).
AUE rarely may be expressed analyticaly by PK parameters:
For the simplest equation C(t)=C0*exp(-k*t) one obtains AUC=C0/k and
AUE=Emax/k * log(1+C0/EC50). Note that even for such a simple case you
cannot calculate AUE based only on AUC value - you need to know also
C0 or k.
For a two-exponential equation (for instance C(t)=A*(exp(-k*t)-exp(-
ka*t)) ) probably there is no closed form for AUE. I tried to
integrate it with Wolfram Mathematica 7.0 and (after several minutes
of hard work!) the integrating routine gave up.
Detailed proof:
Assume all integrals are in the interval from 0 to infinity.
AUC2=Integral C2(t) dt = Integral 0.5*C1(t/2) dt = 0.5*Integral
C1(u)*2 du = Integral C1(u) du = AUC1 { substitution you = t/2 was
used, thus dt = 2 du }
AUE2=Integral C2(t) / [1+C2(t)] dt = Integral 1 / [1 + 1/C2(t)] dt =
Integral 1 / [1+1/( 0.5*C1(t/2) )] dt = Integral { 1 / [1+1/
( 0.5*C1(u) )] }*2 du Integral 2 / [1 + 2/C1(u)] du > Integral 2 / [2
+ 2/C1(u)] du = Integral 1 / [1 + 1/C1(u)] du = AUE1
best regards
Wojciech Jawien
Jagiellonian University
Faculty of Pharmacy
Krakow, Poland
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The following message was posted to: PharmPK
Thanks for your detailed math exercise and explain why there could not
be an unique expression of AUE in terms of AUC, IC50 etc. I
completely agree that AUE is a function of shape of C(t) and not just
AUC. In hindsight, my real interest was to know how AUE is related to
dose in the case that PK is perfectly linear, i.e., the shape of C(t)
does not change. I doubt there is a solution even for that special
case. But the equation you posted with the simplest 1C IV PK curve is
helpful. Thanks!
Jack
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