- On 7 Mar 2000 at 22:44:25, "Bryan Facca" (Bryan.Facca.-a-.metrogr.org) sent the message

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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 - On 8 Mar 2000 at 23:04:01, David_Bourne (david.-a-.boomer.org) sent the message

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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 - On 9 Mar 2000 at 21:51:15, "David Foster" (dmfoster.-at-.u.washington.edu) sent the message

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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 - On 10 Mar 2000 at 10:36:49, ml11439.aaa.goodnet.com sent the message

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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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