- On 25 Nov 1998 at 11:25:35, "Sarawut Oo-puthinan" (sarawuto.-a-.hotmail.com) sent the message

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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 - On 30 Nov 1998 at 12:54:49, ml11439.-a-.goodnet.com (Michael J. Leibold) sent the message

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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 - On 30 Nov 1998 at 12:55:49, "Paul S. Collier" (p.collier.-at-.qub.ac.uk) sent the message

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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/

******************** - On 1 Dec 1998 at 10:46:19, David_Bourne (david.-at-.pharm.cpb.uokhsc.edu) sent the message

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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 - On 2 Dec 1998 at 15:10:47, David_Bourne (david.aaa.pharm.cpb.uokhsc.edu) sent the message

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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/

********************************************** - On 4 Dec 1998 at 11:16:23, David_Bourne (david.-a-.pharm.cpb.uokhsc.edu) sent the message

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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) - On 7 Dec 1998 at 16:46:06, "Mike Jones"(mike.jones.-a-.mdx.com) sent the message

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Richard,

Do you have a reference for the "ProductLog[]" function?

Thanks,

Mike Jones

Micromedex, Inc.

(800) 525-9083 x:6723

Fax: (303) 486-6464 - On 8 Dec 1998 at 12:07:26, Richard Molitor (rmolitor.aaa.u.washington.edu) sent the message

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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/ - On 9 Dec 1998 at 16:00:12, David_Bourne (david.-a-.pharm.cpb.uokhsc.edu) sent the message

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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 - On 10 Dec 1998 at 11:39:59, "Nick Holford" (n.holford.at.auckland.ac.nz) sent the message

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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 - On 11 Dec 1998 at 10:54:29, "Traub, Richard J" (Richard.Traub.at.pnl.gov) sent the message

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 - On 14 Dec 1998 at 12:06:13, David_Bourne (david.aaa.pharm.cpb.uokhsc.edu) sent the message

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[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

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