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Dear sir, all pharmacokineticists
I want to simulate the curve between concentration and time of drug
with non-linear kinetics by computer.
I can't integrate the Mechaelis-Menten equation into function of
concentration. I can't draw the curve from this equation:
Vmax (t-t0) = C0-C + Km.ln C0/C.
If we give drug in repeated doses. We can derive the equations of
conc.and time for multiple dose administration, or not?
Can anybody help me?
Thank you in advance
Sarawut Oo-puthinan
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Dear Stuart,
I have integrated the Michaelis-Menten expression for oral and
itravenous doses many times using numerical integration. Specifically,
Runge-Kuta numerical integration as described in various numerical analysis
textbooks. The Runge-Kuta equation may be programmed into a computer
spreadsheet, or into a programmable calculator. Alternatively, the Boomer
program supplied by Dr.Bourne also has a Runge-Kuta numerical analysis
program, although I have never used it myself.
There should be several textbooks on numerical anlaysis available in any
library, public or university, or in some large bookstores.
Of course, the integrated form of the Michaelis-Menten equation is not
used, but rather the differential equation.
Mike Leibold, PharmD
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Dear Sarawut Oo-puthinan
Check the following reference re your recent query and see if
this is of any help to you.
Collier, P.S. (1983). A calculator/microcomputer program for solving
the Michaelis-Menten equation in terms of concentration. Drug Intell.
and Clin. Pharmacy, 17, 559-560.
Paul S. Collier
*********************
School of Pharmacy
The Queen's University of Belfast
97 Lisburn Road
Belfast BT9 7BL
N. Ireland, U.K
Tel: +44 (0)1232 272009
FAX: +44 (0)1232 247794
Email: p.collier.aaa.qub.ac.uk
http://www.qub.ac.uk/pha/index.html
**********************
Editor for Europe
Biopharmaceutics & Drug Disposition
http://www.interscience.wiley.com/jpages/0142-2782/
********************
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[Two solutions - db]
Date: Mon, 30 Nov 1998 12:56:03 -0700
From: "David Nix, Pharm D."
Organization: College of Pharmacy
MIME-Version: 1.0
To: PharmPK.aaa.pharm.cpb.uokhsc.edu
Subject: Re: PharmPK Equations of conc. vs time for non-linear kinetics
It is true that you cannot solve for Concentration as a function of time
with non-linear (MM) pharmacokinetics. You will need to use numerical
integration and the differential equations. Several programs have built
in numerical integrators and would be able to perform simulations. I
personally use Adapt II for this purpose.
David Nix
---
Date: Tue, 01 Dec 1998 08:30:24 -0800
From: "Traub, Richard J"
Subject: RE: PharmPK Equations of conc. vs time for non-linear kinetics
To: "'PharmPK.aaa.pharm.cpb.uokhsc.edu'"
MIME-version: 1.0
Sarawut Oo-puthinan says:
I want to simulate the curve between concentration and time of
drug
with non-linear kinetics by computer.
I can't integrate the Mechaelis-Menten equation into function of
concentration.
Can anybody help me?
+++++++++++
As I understand the question, Sarawut would like to solve equation
1 for
C[t].
dC[t]/dt = - Vm * C[t] / (Km + C[t] ) eq 1
According to MMA 3.0 the solution is equation 2
C[t] = Km * ProductLog[ exp[ -(t * Vm / Km ) + Ln[ C[0] * exp[ C[0] /
Km] ] ] / Km], eq2
where C[0] is the concentration at time t=0, C[t] is the concentration
at time t=t, Ln[.] is the natural log, exp[.] is the exponential function, and
ProductLog[z] gives the principal solution for w in z=w*exp(w). For z > -1/e,
ProductLog[z] is real.
The only problem with equation 2 is the ProductLog[] is not an
especially common function and, depending on the value of z, naive
implementations will sometimes fail to converge or return a complex value
when a
real value is correct.
Hope this helps
Richard J. Traub
Pacific Northwest National Laboratory
MSIN K3-55
P.O. Box 999
Richland, WA 99352
(509) 375-4385 - Voice
(509) 375-2019 - FAX
E mailto:richard.traub.-at-.pnl.gov
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[Two more replies - db]
X-Sender: walt.-at-.mail.simulations-plus.com
Date: Tue, 01 Dec 1998 14:27:00 -0800
To: PharmPK.aaa.pharm.cpb.uokhsc.edu
From: Walt Woltosz
Subject: Re: PharmPK Re: Equations of conc. vs time for non-linear
kinetics
Cc: bolger.-at-.simulations-plus.com, ronc.-a-.simulations-plus.com
Mime-Version: 1.0
You might want to consider GastroPlus(TM), a new simulation package from
Simulations Plus, Inc. that is a numerical integration of differential
equations for transit, release, dissolution, and absorption, as well as
one- or two-compartment pharmacokinetics. There is an Academic Demo version
(cheap but very limited) which can be used for teaching purposes, and an
Academic Research version which can be used for real research.
The model is based on the Compartmental Absorption and Transit (CAT) model
after Amidon and Yu, but enhanced in many ways.
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
* Walt Woltosz Phone: (805) 723-7723 *
* Chairman & CEO FAX: (805) 723-5524 *
* Simulations Plus, Inc. (NASDAQ:SIMU) *
* 1220 West avenue J *
* Lancaster, CA 93534-2902 *
* U.S.A. *
* http://www.simulations-plus.com *
* walt.at.simulations-plus.com *
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
---
From: "Paul S. Collier"
Sender: P.Collier.-at-.Queens-Belfast.AC.UK
Reply-To: p.collier.-a-.qub.ac.uk
To: PharmPK.-a-.pharm.cpb.uokhsc.edu
Subject: PharmPK Re: Equations of conc. vs time for non-linear kinetics
Date: Wed, 02 Dec 1998 09:19:13 +0000
Priority: NORMAL
X-Authentication: IMSP
MIME-Version: 1.0
Subject: Re: PharmPK Equations of conc. vs time for non-linear kinetics
It is incorrect to say that one cannot solve for Concentration as a
function of time
with non-linear (MM) pharmacokinetics. The following references
discuss this situation and give programmes (written in BASIC and in
FORTRAN) that solve the equation of concentration as a function of time.
Programmable Solution For The Michaelis-Menten Equation
Collier PS
Drug Intelligence & Clinical Pharmacy, 1983, Vol.17, No.7-8, Pp.559-560
Computation Of The Explicit Solution To The Michaelis-Menten Equation
Beal SL
Journal Of Pharmacokinetics And Biopharmaceutics, 1983, Vol.11, No.6,
pp.641-657
On The Solution To The Michaelis-Menten Equation
Beal SL
Journal Of Pharmacokinetics And Biopharmaceutics, 1982, Vol.10, No.1,
Pp.109-119
Paul S. Collier
**********************************************
School of Pharmacy
The Queen's University of Belfast
97 Lisburn Road
Belfast BT9 7BL
N. Ireland, U.K
Tel: +44 (0)1232 272009
FAX: +44 (0)1232 247794
Email: p.collier.-at-.qub.ac.uk
http://www.qub.ac.uk/pha/index.html
**********************************************
Editor for Europe
Biopharmaceutics & Drug Disposition
http://www.interscience.wiley.com/jpages/0142-2782/
**********************************************
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[Two replies - db]
X-Sender: jelliffe.-at-.hsc.usc.edu
Date: Wed, 02 Dec 1998 17:54:46 -0800
To: PharmPK.at.pharm.cpb.uokhsc.edu
From: Roger_Jelliffe
Subject: Re: PharmPK Re: Equations of conc. vs time for non-linear
Mime-Version: 1.0
Dear All:
About MM equations, etc., you can make a MM (or any other linear or
nonlinear structural model) and make an NPEM population model if you wish.
You can use our BOXES program to place boxes (compartments) on the screen
and connect them with different kinds of arrows for pathways (linear, MM,
either with stated descriptors (covariates) etc., to make the model, or you
can type in the differential equations for any model you want. You then use
our NPEM software for the PC to prepare and send (using SSH, a secure
version of Telnet) this model, along with the subject data sets (entered in
the standard USC*PACK clinical file format) to the Cray T3E at the San
Diego Supercomputer Center. Do the analysis, and send the results back to
the PC to see them. The parallel software on the Cray greatly speeds up the
analysis so the job is done fairly quickly for having to use a DE solver
and perform the integrations. It is a useful option for someone who wants
to make nonlinear PK/PD models.
Further, the strength of nonparametric population modeling is that
you get
not only the means, variances, etc., but also the entire joint probability
density of the parameters. This means that when you wish to actually use
(not just report) use this information, you now have a tool to evaluate the
precision with which any dosage regimen will fail to achieve any desired
target goal. This is not possible with parametric models, which have only a
single value for each model parameter.
Hope to hear from anyone who is interested.
Sincerely,
Roger Jelliffe
Roger W. Jelliffe, M.D.
USC Lab of Applied Pharmacokinetics
(323)442-1300, fax (323)442-1302
jelliffe.-a-.hsc.usc.edu
************************************
You might consider looking at our web site for info on new
software, technical reports, and workshops
Our web address is www.usc.edu/hsc/lab_apk
************************************
---
Date: Thu, 03 Dec 1998 09:32:36 -0800
From: "Paul B. Laub"
Organization: Incyte Pharmaceuticals
MIME-Version: 1.0
To: PharmPK.-at-.pharm.cpb.uokhsc.edu
Subject: Re: PharmPK Equations of conc. vs time for non-linear kinetics
Dear Sarawut,
For PK and PD simulations, an excellent package is Adapt II, which is available
free or for nominal cost from Prof. David Z. D'Argenio, Biomedical Simulations
Resource, University of Southern California. The BMSR has a web site, I
believe. Be aware that you will need a FORTRAN 77 compiler to use it, though
the user need not be a programmer to find it useful. Adapt II does simulation
(like you want), fitting, and sampling calculations. It is flexible enough to
handle nonlinear kinetics and multiple dosing.
Paul Laub
scientific programmer, Incyte Pharma., Inc.
--
Paul B. Laub Incyte Pharmaceuticals 3174 Porter Dr. Palo Alto, CA 94304
(650) 845-5411 (voice) plaub.at.incyte.com (650) 855-0572 (fax)
Back to the Top
Richard,
Do you have a reference for the "ProductLog[]" function?
Thanks,
Mike Jones
Micromedex, Inc.
(800) 525-9083 x:6723
Fax: (303) 486-6464
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The ProductLog is the multi-valued function that inverts y==x*Exp[x] (it
is sometimes called Lambert's W function in the literature).
Hope this helps.
Richard Molitor, R.Ph.
http://www.angelfire.com/wa/pharmacist/
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[Two more replies - db]
Date: Tue, 08 Dec 1998 16:49:56 -0800
From: "Traub, Richard J"
Subject: RE: PharmPK Re: Equations of conc. vs time for non-linear ki
To: "'PharmPK.-a-.pharm.cpb.uokhsc.edu'"
MIME-version: 1.0
Also, The ProductLog[] function is implemented in Mathematica Version
3.0 and later. (I don't know about earlier versions of Mathematica.)
You can learn more about Lambert's W function at:
http://www.astro.virginia.edu/~eww6n/math/math0.html
Note: Because the information at this URL has been published as a book
(see the info at the URL) the author has to block access to some pages. The
blocked pages change daily so if at first you don't succeed, ...
The implementation of the ProductLog function is tricky. Naive
implementations will not always give the correct answer. So far, I have not
found an implementation outside of Mathematica. If anyone finds an
implementation, please let me know. If I find a good implementation, I'll
forward it to the list.
Hope this helps.
Richard J. Traub
Pacific Northwest National Laboratory
MSIN K3-55
P.O. Box 999
Richland, WA 99352
(509) 375-4385 - Voice
(509) 375-2019 - FAX
E mailto:richard.traub.at.pnl.gov
---
From: david_william.tudor.aaa.pharma.Novartis.com
X-Lotus-FromDomain: PH.-at-.N1
To: PharmPK.aaa.pharm.cpb.uokhsc.edu
Date: Wed, 9 Dec 1998 11:21:52 +0100
Subject: Re: PharmPK Re: Equations of conc. vs time for non-linear ki
Mime-Version: 1.0
In reference to the "ProductLog" function: Something must be missing in
the definition given (" multi-valued function that inverts y==x*exp(x) ").
The function y==x*exp(x) is one-to-one and monotone, hence invertible and
its inverse is also a function (meaning single-valued).
What's missing?
Thanks,
David Tudor
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Richard,
You have wetted my appetite but I cannot find out about Lamberts W at the
moment because of the copyright block. Actually I would be more interested
in knowing what good this function is in a pharmacokinetic context. Can you
provide an example?
Nick
--
Nick Holford, Dept Neurology,L226
OHSU,3181 SW Sam Jackson Park Rd,Portland,OR 97201,USA
email:n.holford.-a-.auckland.ac.nz tel:+1(503)494-7228 fax:494-7242
http://www.phm.auckland.ac.nz/Staff/NHolford/nholford.htm
Back to the Top
When speaking of the ProductLog[] and Lambert's W, Nick Holford asks:
Actually I would be more interested in knowing what good this function
is in a pharmacokinetic context. Can you
provide an example?
++++++++
This is a good example of a case where the tread has taken on a life of
it's own and we all forgot how we got here.
The original question was something like:
How do I solve the MM eqn (Eq 1) for C[t] analytically? i.e., without
needing a differential equation solver.
dC[t]/dt = - Vm * C[t] / (Km + C[t] ) eq 1
I used the DSolve[] function of Mathematica 3.0 to derive Eq. 2
C[t] = Km * ProductLog[ exp[ -(t * Vm / Km ) + Ln[ C[0] * exp[
C[0] / Km] ] ] / Km], eq2
where C[0] is the concentration at time t=0, C[t] is the
concentration
at time t=t, Ln[.] is the natural log, exp[.] is the exponential
function, and
ProductLog[z] gives the principal solution for w in z=w*exp(w). For z
> -1/e,
ProductLog[z] is real.
Using Eq 2 you can compute the concentration at any time given the
original concentration and Km and Vm.
As it turns out, ProductLog[] is Wolfram Research's name for the
solution to Lambert's W function.
To answer Nick's question, the ProductLog[] function or something else
that solves Lambert's W is necessary if you want to implement equation 2 on
your
computer. Off the top of my head, I can't think of any other pharmacokinetic
use for the function.
David Tudor asked about why the function was called multivalued. The
reason is because if you write out the solution for w, i.e.
w = log[z] - log[w], where z is the input and you want to find w
then for values of z less than e (about 2.7) naive iterative solutions
for w will converge to a complex number rather than a real number. (try it and
see) A real number is what you want.
It turns out that a freely available implementation of the solution to
Lambert's W is available from netlib as "wapr" or TOMS Algorithm 743.
Check out the NIST Guide to Available Mathematical Software at
http://gams.cam.nist.gov/
Hope this helps
Richard J. Traub
Pacific Northwest National Laboratory
MSIN K3-55
P.O. Box 999
Richland, WA 99352
(509) 375-4385 - Voice
(509) 375-2019 - FAX
E mailto:richard.traub.aaa.pnl.gov
Back to the Top
[Two replies -db]
From: "Nick Holford"
To:
Subject: Re: PharmPK Re: Equations of conc. vs time for non-linear ki
Date: Sun, 13 Dec 1998 21:13:44 -0800
MIME-Version: 1.0
X-Priority: 3
Richard,
Thanks for your helpful reply. I had indeed missed part of the thread.
> To answer Nick's question, the ProductLog[] function or something else
>that solves Lambert's W is necessary if you want to implement equation 2 on
>your
>computer.
Not exactly *necessary*. Solving differential equations is often a more
flexible way to deal with this kind of model and many other complex PK
problems.
Nick
---
X-Sender: rjantzen.aaa.email.vill.edu (Unverified)
Date: Fri, 11 Dec 1998 15:59:21 -0500
To: richard.traub.aaa.pnl.gov, PharmPK.at.pharm.cpb.uokhsc.edu, david.-at-.boomer.org
From: Robert Jantzen
Cc: asarkahian.at.magainin.com, rjantzen.at.email.vill.edu
Mime-Version: 1.0
Dear Pharmacokinetics people,
My wife is in the pharmapk industry and subscribes to this email list which
just had a blurb about what good the Lambert W function is for this field.
[PharmPK Re: Equations of conc. vs time for non-linear ki...]
Just by coincidence, this past year, the Lambert W function arose in a
pharmpk problem with a friend of ours in a small research company in a way
that might be of amusement to you.
It is described in the web page:
http://renoir.vill.edu/math/archives/maple/misc/lambert/lamb.htm
This is the HTML output of a MAPLE Release 5 worksheet. MAPLE and
Mathematica are head to head competitors: MAPLE developed in a university
setting by education oriented people, Mathematica developed by a smart guy
who acquired an army of private industry help. Mathematica is still
superior in graphics, and maybe in certain advanced applications, but MAPLE
seems to be better in keeping with standard mathematical symbols and style
of command structure and is easier for the beginning user than Mathematica,
which seems to need an expert consultant as soon as you want to do
something sophistocated. I know, I use both. Of course no software is perfect.
Like my worksheet which I just dashed off for my own amusement, thinking no
one else would ever be interested. Let me know what you think. By the way,
you can drag the vertical frame separator all the way to the left with your
mouse to make the viewing window of the worksheet as wide as possible.
bob jantzen
dept of mathematical sciences
villanova university
villanova pa 19085 usa
http://renoir.vill.edu/faculty/jantzen/html/home.html
PharmPK Discussion List Archive Index page
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