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Dear All,
I came across a model described in an abstract as "triexponential".
Could someone show this as a math expression and give an example ?
Preferably with a drug in human use.
Many Thanks
Bryan Facca RPh PharmD
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[Four replies - db]
X-Sender: dfarrier.-at-.mail.bright.net
Date: Wed, 08 Mar 2000 01:23:59 -0500
To: PharmPK.-a-.boomer.org
From: "David S. Farrier"
Subject: Re: PharmPK Triexponential - Model example
Bryan,
The term "triexponential" means that a blood level curve appears to be
described by the sum of three exponential terms according to the expression:
concentration = A*exp(-at) + B*exp(-bt) + C*exp(-ct)
where t is time and the final term could be positive or negative
to reflect iv or extravascular doses, respectively
To see an example of how each of these terms is obtained using the method
of feathering or curve stripping, download a demo of PK Solutions from
http://www.SummitPK.com
You can also download a free compilation of equations showing how such
exponential terms are used to calculate a variety of useful pharmacokinetic
parameters.
Regards,
David
David S. Farrier, Ph.D.
Summit Research Services
1374 Hillcrest Drive
Ashland, OH 44805 USA
Tel/Fax: (419)-289-9207
Email: DFarrier.-a-.SummitPK.com
Web Site: www.SummitPK.com
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X-Sender: mentor.aaa.hardlink.com
Date: Wed, 08 Mar 2000 01:35:21 -0500
To: PharmPK.-at-.boomer.org
From: Daro Gross
Subject: Re: PharmPK Triexponential - Model example
I have never heard of this terminology before, but take a look at the
following equation and see if this makes sense to you:
expected result = a1 * e (expr) + a2 * e (expr) + a3 * e (expr)
or
expected result = a1 a2 a3 * exp (x1) 0 0
b1 b2 b3 0 exp(y1) 0
c1 c2 c3 0 0 exp(z1)
The expected result would be mapped into a linear 3D vector space described
by linearly independent exponential curves, i.e., use of three drugs having
with linearly independent exponential responses would map into a 3D space
for which one could test for linear dependence and filter out "data" by
testing for linear dependence.
Most PK/PD models assume there to be dependence of some kind between the
variables, hence this would be a means of creating a data filter for
identifying unexpected dependencies between the variables. This is just a
means of working with variables most easily described in terms of
exponential response, which is actually the the norm for most variables in
a feed-back loop, i.e., most of medicine.
The mathematics can get tricky, but one can often perform more accurate and
useful modeling of PK/PD studies using exponential response and
constructing filters to either identify dependencies or filter out known
independencies.
There are few true linear response curves to be found in medicine---it just
turns out that linear models are faster and easier to work with in many
cases. If one has the time, try this type of modeling, but treating
patients often leaves little time for contructing such complex models.
----Daro Gross
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Date: Wed, 8 Mar 2000 02:18:14 -0700 (MST)
X-Sender: ml11439.-a-.pop.goodnet.com
To: PharmPK.aaa.boomer.org
From: ml11439.-at-.goodnet.com (Michael J. Leibold)
Subject: Re: PharmPK Triexponential - Model example
Bryan,
Triexponential refers to the presence of three slopes in the
log plasma concentration versus time curve.
Cp = Ae-at + Be-bt + Ge-gt
In linear compartmental pharmacokinetics this is interpreted as a
three-compartment model:
Xo
\
k12 k13
Cpt2<------->Cpt1<------->Cpt3
k21 \ k31
\->k10
This is a linear mammillary model with elimination from a central
compartment wich usually represents the plasma compartment. The system
is described by linear first order differential equations describing
the change in compartmental amounts of drug with time:
dX1/dt= -(k13+k10+k12)X1 + k21X2 +k31X3
dX2/dt= k12X1 + -k21X2
dX3/dt= k12X1 -k31X3
This system of differential equations can be solved by Laplace
transforms and matrix algebra. The matix representation of the Laplace
transformed system of differential equations is:
[SI-A][Xs]= [Us]
Where the Laplace transform of the system of differential equations
above is equal to the Laplace transformed vector of dose input, for
example: (Ko/s)(1-e-Ts).
[(s+k13+k10+k12) -k21 -k31][X1s] [(Ko/s)(1-e-Ts)]
[ -k12 (s+k21) 0 ][X2s] = [ 0 ]
[ -k13 0 (s+k31)][X3s] [ 0 ]
This can be solved by matrix algebra to yield Laplace transformed
quantities of each compartment, most important of which is the central
compartment quantity (X1s):
X1s= [(Ko/s)(1-e-Ts)(s+k21)(s+k31)]/[(s+a)(s+b)(s+g)]
The inverse Laplace transform of the above results in an equation
describing the amount in the central compartment as function of time,
and dividing this by the Vc (volume of the central compartment) results
in the equation for the plasma concentration over time:
Cp= Ko(k21-a)(k31-a)(1-e-aT)(e-at')/[a(b-a)(g-a)Vc] +
Ko(k21-b)(k31-b)(1-e-bT)(e-bt')/[b(a-b)(g-b)Vc] +
Ko(k21-g)(k31-g)(1-e-gT)(e-gt')/[g(a-g)(b-g)Vc]
The above equation describes the triexponential plasma concentration
curve of a drug being administered by a an intermittent infusion, where
T= infusion time and t'= time after infusion.
The multiple dose form of the above equation is:
Cp= Ko(k21-a)(k31-a)(1-e-aT)(1-e-naTau)(e-at')/[a(b-a)(g-a)Vc(1-e-aTau)] +
Ko(k21-b)(k31-b)(1-e-bT)(1-e-nbTau)(e-bt')/[b(a-b)(g-b)Vc(1-e-bTau)] +
Ko(k21-g)(k31-g)(1-e-gT)(1-e-ngTau)(e-gt')/[g(a-g)(b-g)Vc(1-e-gTau)]
Examples of drugs which exhibit triexpoential plasma concentration
curves are vancomycin and aminoglycosides. However, this is difficult to
detect as the curves really appear two compartment. Vancomycin has an
initial rapid distribution phase (T1/2~=7min) which can be detected with
numerous plasma samples and careful analysis. Aminglycosides have a long
washout phase due to renal tissue binding and release which can be detected
with plasma samples taken long after the drug has been administered to
detect a slow decline (T1/2~=100-200 hours) in plasma concentrations as drug
is released from renal tissue.
Mike Leibold, PharmD, RPh
ML11439.-a-.goodnet.com
References
1) Gibaldi, M., Perrier, D., Pharmacokinetics, New York, Marcel Dekker
1975
2) Gibaldi, M., Perrier, D., Pharmacokinetics 2nd ed, New York, Marcel Dekker
1982
3) Schumacher, G.E., Therapeutic Drug Monitoring, Norwalk, Appleton&
Lange 1995
4) Evans, W.E., Schentag, J.J., Jusko, W.J., Applied Pharmacokinetics 3rd ed,
Vancouver, Applied Therapeutics 1992
5) Wagner, J., Fundamentals of Clinical Pharmacokinetics, Hamilton, Drug
Intelligence Publications 1975
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Date: Wed, 08 Mar 2000 16:03:19 -0600
From: "Vahn Lewis"
X-Accept-Language: en
To: PharmPK.aaa.boomer.org
Subject: Re: PharmPK Triexponential - Model example
Propofol has been described by a triexponential elimination.
It would look something like:
Cp=Ae-at+Be-bt+Ce-ct
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The tri-exponential is simply a sum of three exponentials. While it
normally is used to describe plasma decay of, for example, drug
concentration following an iv bolus, it can also be used in other
circumstances. When dealing with systems of first order, linear constant
coefficient differential equations (i.e. the simplest kind of compartmental
model), it means your system contains three compartments.
I suggest you look at the SAAM II software system which deals with sums of
exponentials in its numerical application, and compartmental models in its
compartmental application. You can find more at:
http://www.saam.com
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Discussion group,
I would like to report an error in my email regarding the
three-compartment system of differential equations. The first
microconstant in the differential equation for X3 should be
k13 and not k12. That is, the system of differential equations
should be:
dX1/dt= -(k13+k10+k12)X1 + k21X2 +k31X3
dX2/dt= k12X1 + -k21X2
dX3/dt= k13X1 -k31X3
However, this was corrected later in the email when the
Laplace transformed matrix representation is discussed.
Mike Leibold, PharmD, RPh
ML11439.-a-.goodnet.com
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