# PharmPK Discussion - Triexponential - Model example

PharmPK Discussion List Archive Index page
• On 7 Mar 2000 at 22:44:25, "Bryan Facca" (Bryan.Facca.-a-.metrogr.org) sent the message
`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 ThanksBryan Facca RPh PharmD`
Back to the Top

• On 8 Mar 2000 at 23:04:01, David_Bourne (david.-a-.boomer.org) sent the message
`[Four replies - db]X-Sender: dfarrier.-at-.mail.bright.netDate: Wed, 08 Mar 2000 01:23:59 -0500To: PharmPK.-a-.boomer.orgFrom: "David S. Farrier" Subject: Re: PharmPK Triexponential  - Model exampleBryan,The term "triexponential" means that a blood level curve appears to bedescribed 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, respectivelyTo see an example of how each of these terms is obtained using the methodof feathering or curve stripping, download a demo of PK Solutions fromhttp://www.SummitPK.comYou can also download a free compilation of equations showing how suchexponential terms are used to calculate a variety of useful pharmacokineticparameters.Regards,DavidDavid S. Farrier, Ph.D.Summit Research Services1374 Hillcrest DriveAshland, OH 44805 USATel/Fax:  (419)-289-9207Email:     DFarrier.-a-.SummitPK.comWeb Site: www.SummitPK.com---X-Sender: mentor.aaa.hardlink.comDate: Wed, 08 Mar 2000 01:35:21 -0500To: PharmPK.-at-.boomer.orgFrom: Daro Gross Subject: Re: PharmPK Triexponential  - Model exampleI have never heard of this terminology before, but take a look at thefollowing equation and see if this makes sense to you:expected result = a1 * e (expr) + a2 * e (expr) + a3 * e (expr)orexpected 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 describedby linearly independent exponential curves, i.e., use of three drugs havingwith linearly independent exponential responses would map into a 3D spacefor which one could test for linear dependence and filter out "data" bytesting for linear dependence.Most PK/PD models assume there to be dependence of some kind between thevariables, hence this would be a means of creating a data filter foridentifying unexpected dependencies between the variables. This is just ameans of working with variables most easily described in terms ofexponential response, which is actually the the norm for most variables ina feed-back loop, i.e., most of medicine.The mathematics can get tricky, but one can often perform more accurate anduseful modeling of PK/PD studies using exponential response andconstructing filters to either identify dependencies or filter out knownindependencies.There are few true linear response curves to be found in medicine---it justturns out that linear models are faster and easier to work with in manycases. If one has the time, try this type of modeling, but treatingpatients often leaves little time for contructing such complex models.----Daro Gross---Date: Wed, 8 Mar 2000 02:18:14 -0700 (MST)X-Sender: ml11439.-a-.pop.goodnet.comTo: PharmPK.aaa.boomer.orgFrom: ml11439.-at-.goodnet.com (Michael J. Leibold)Subject: Re: PharmPK Triexponential  - Model exampleBryan,     Triexponential refers to the presence of three slopes in thelog plasma concentration versus time curve.     Cp = Ae-at + Be-bt + Ge-gt     In linear compartmental pharmacokinetics this is interpreted as athree-compartment model:                          Xo                           \                     k12           k13              Cpt2<------->Cpt1<------->Cpt3                     k21    \      k31                             \->k10     This is a linear mammillary model with elimination from a centralcompartment wich usually represents the plasma compartment. The systemis described by linear first order differential equations describingthe 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 Laplacetransforms and matrix algebra. The matix representation of the Laplacetransformed system of differential equations is:                    [SI-A][Xs]= [Us]     Where the Laplace transform of the system of differential equationsabove is equal to the Laplace transformed vector of dose input, forexample: (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 transformedquantities of each compartment, most important of which is the centralcompartment 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 equationdescribing the amount in the central compartment as function of time,and dividing this by the Vc (volume of the central compartment) resultsin 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 concentrationcurve of a drug being administered by a an intermittent infusion, whereT= 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 concentrationcurves are vancomycin and aminoglycosides. However, this is difficult todetect as the curves really appear two compartment. Vancomycin has aninitial rapid distribution phase (T1/2~=7min) which can be detected withnumerous plasma samples and careful analysis. Aminglycosides have a longwashout phase due to renal tissue binding and release which can be detectedwith plasma samples taken long after the drug has been administered todetect a slow decline (T1/2~=100-200 hours) in plasma concentrations as drugis released from renal tissue.                    Mike Leibold, PharmD, RPh                    ML11439.-a-.goodnet.comReferences1) Gibaldi, M., Perrier, D., Pharmacokinetics, New York, Marcel Dekker    19752) Gibaldi, M., Perrier, D., Pharmacokinetics 2nd ed, New York, Marcel Dekker    19823) Schumacher, G.E., Therapeutic Drug Monitoring, Norwalk, Appleton&    Lange 19954) Evans, W.E., Schentag, J.J., Jusko, W.J., Applied Pharmacokinetics 3rd ed,    Vancouver, Applied Therapeutics 19925) Wagner, J., Fundamentals of Clinical Pharmacokinetics, Hamilton, Drug    Intelligence Publications 1975---Date: Wed, 08 Mar 2000 16:03:19 -0600From: "Vahn Lewis" X-Accept-Language: enTo: PharmPK.aaa.boomer.orgSubject: Re: PharmPK Triexponential  - Model examplePropofol has been described by a triexponential elimination.It would look something like:Cp=Ae-at+Be-bt+Ce-ct`
Back to the Top

• On 9 Mar 2000 at 21:51:15, "David Foster" (dmfoster.-at-.u.washington.edu) sent the message
`The tri-exponential is simply a sum of three exponentials.  While itnormally is used to describe plasma decay of, for example, drugconcentration following an iv bolus, it can also be used in othercircumstances.  When dealing with systems of first order, linear constantcoefficient differential equations (i.e. the simplest kind of compartmentalmodel), it means your system contains three compartments.I suggest you look at the SAAM II software system which deals with sums ofexponentials in its numerical application, and compartmental models in itscompartmental application.  You can find more at:http://www.saam.com`
Back to the Top

• On 10 Mar 2000 at 10:36:49, ml11439.aaa.goodnet.com sent the message
`Discussion group,     I would like to report an error in my email regarding thethree-compartment system of differential equations. The firstmicroconstant in the differential equation for X3 should bek13 and not k12. That is, the system of differential equationsshould 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 theLaplace transformed matrix representation is discussed.                   Mike Leibold, PharmD, RPh                   ML11439.-a-.goodnet.com`
Back to the Top

Want to post a follow-up message on this topic? If this link does not work with your browser send a follow-up message to PharmPK@boomer.org with "Triexponential - Model example" as the subject

Copyright 1995-2010 David W. A. Bourne (david@boomer.org)