- On 9 Aug 1999 at 21:52:51, "Thierry.Buclin" (Thierry.Buclin.-a-.chuv.hospvd.ch) sent the message

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Dear Colleagues,

I have to build up simulations for complicated compartmental models (e.g.

drug with 3-compartment disposition and generating N metabolites with

partial interconversion, etc.). It would be both more convenient and more

exact to use integrated equations and macroconstants for that. If all

transfer processes are linear, this is theoretically possible for any

number of compartments. But the derivation of such equations from

microconstants becomes incredibly time-consuming and produces huge

expressions difficult to handle with paper and pencil when you deal with

many-compartment models.

So does anybody know about a software which would perform symbolic

integration and provide integrated equations for a multicompartmental model

specified in microconstants, without regard to the amount of arrows and

boxes it includes ? Thank you in advance

Thierry BUCLIN, MD

Division of Clinical Pharmacology

University Hospital CHUV - Beaumont 633

CH 1011 Lausanne - SWITZERLAND

Tel: +41 21 314 42 61 - Fax: +41 21 314 42 66 - On 10 Aug 1999 at 22:43:16, David_Bourne (david.aaa.boomer.org) sent the message

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[A few replies - Mathematica, MATLAB, and Maple for symbolic

integration. ADAPT II and others SAAM II, WinNONLIN, even Boomer ;-)

to perform numerical integration - see

http://www.boomer.org/pkin/soft.html for info re programs to do

numerical integration. Maybe I need to add entries for the symbolic

integrators - db]

Reply-To: "Stephen Duffull"

From: "Stephen Duffull"

To:

Subject: Re: PharmPK Integration of compartmental models

Date: Tue, 10 Aug 1999 09:18:49 +0100

X-Priority: 3

Thiery:

I have used Mathematica for this and it works fairly well.

I understand MATLAB (& Maple) can also perform symbolic

integration - although I have not used either for this

purpose. Please note that I say Mathematica works "fairly

well", I have had problems when integrating some systems of

linear ODEs. Indeed in one case I found it less

problematical to integrate them by hand using Laplace

transforms.

Regards

Steve

=====================

Stephen Duffull

School of Pharmacy

University of Manchester

Manchester, M13 9PL, UK

Ph +44 161 275 2355

Fax +44 161 275 2396

---

From: "Bachman, William"

To: "'PharmPK.-at-.boomer.org'"

Subject: RE: PharmPK Integration of compartmental models

Date: Tue, 10 Aug 1999 07:54:39 -0400

One such program (to provide analytical solutions to differential equation

models) is Mathematica. There are others including Maple).

William J. Bachman, Ph.D.

GloboMax LLC

Senior Scientist

7250 Parkway Drive, Suite 430

Hanover, MD 21076

Voice (410) 782-2212

FAX (410) 712-0737

bachmanw.aaa.globomax.com

---

Reply-To:

Sender: "Eric Masson"

To:

Subject: RE: PharmPK Integration of compartmental models

Date: Tue, 10 Aug 1999 09:11:07 -0400

X-Priority: 3 (Normal)

Importance: Normal

I would try to fit this model using differential equations. A software such

as ADAPTII will let you do that.

Eric Masson, Pharm.D.

Scientific Director,

Anapharm inc

2050, boul Rene-Levesque West,

Ste-Foy, QC, Canada, G1V-2K8

418-527-4000 (EXT:222)

FAX: 418-527-3456

---

X-Sender: jelliffe.aaa.hsc.usc.edu

Date: Tue, 10 Aug 1999 13:40:12 -0700

To: PharmPK.at.boomer.org

From: Roger Jelliffe

Subject: Re: PharmPK Integration of compartmental models

Dear Thierry:

The ADAPT II programs of D'Argenio and Schumitzky use differential

equations and ODE solvers to do such simulations for both linear and

nonlinear models. You might consider them. To my knowledge, they do not do

symbolic integration, but use well-tested ODE solvers to do the job. They

have been around for many years, are well tested, and have a very good

reputation. Dr. D'Argenio's email is

dargenio.-at-.bmsrs.usc.edu

Hope this is useful to you.

Roger Jelliffe

--- - On 11 Aug 1999 at 21:49:45, "C. Anthony Hunt" (hunt.-a-.itsa.ucsf.edu) sent the message

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Thierry, for complex simulations I recommend:

http://www.VISSIM.com/PRODUCTS/pro-ov.html

http://www.mathworks.com/products/simulink/ and/or

http://www.hps-inc.com/products/stella/chapter_one/chapone.html

For symbolic integration the best is

http://www.mathematica.com/solutions/biomed/

for more ideas start with:

http://www.biosoft.com/main.htm

-Tony Hunt- - On 11 Aug 1999 at 23:05:36, "Traub, Richard J" (Richard.Traub.at.pnl.gov) sent the message

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I use Mathematica(TM) for this sort of problem.

Mathematica isn't as

easy to use as one might like but it does work. You can get information

concerning Mathematica at the Wolfram web site http://www.wolfram.com . Hope

this helps

Richard J. Traub

MSIN K3-55

Pacific Northwest National Laboratory

P.O. Box 999

Richland, WA 99352

(509) 375-4385 (voice)

(509) 375-2019 (FAX)

mailto:richard.traub.-at-.pnl.gov - On 12 Aug 1999 at 23:04:04, "David Foster" (dmfoster.-a-.u.washington.edu) sent the message

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Dear Thierry,

presently available techniques of differential equation integration make

it unnecessary to explicitly integrate compartmental model equations to

obtain sums of exponentials. Plus (as you suggest), these expressions

become so cumbersome for more than three compartments that they quickly

outgrow their usefulness.

A tool that accomplishes what you suggest is SAAM II. In SAAM II, the

compartmental model can be built graphically on the screen, and the

differential equations (however complicated) are handled by the program.

Thus, you can simulate the output of any model you can conceive, for any

value of the (micro)parameters.

More information about SAAM II is available at: http://www.saam.com or

by emailing to info.aaa.saam.com. A demo version of the program is also

available for you to try.

Best wishes,

David M. Foster

*******************

Please note the change of my e-mail address to:

dmfoster.at.u.washington.edu

****************** - On 13 Aug 1999 at 13:23:13, "Bachman, William" (bachmanw.at.globomax.com) sent the message

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Dear David

There are still valid reasons for having analytical solutions to

differential equation models. Speed for one. Particularly in the

population context. Using the analytical solution for population analyses

with large populations can save an ENORMOUS amount of computing time

compared to the diff. eqn.

Best regards,

Bill

William J. Bachman, Ph.D.

GloboMax LLC

Senior Scientist

7250 Parkway Drive, Suite 430

Hanover, MD 21076

Voice (410) 782-2212

FAX (410) 712-0737

bachmanw.aaa.globomax.com - On 14 Aug 1999 at 22:25:20, David_Bourne (david.at.boomer.org) sent the message

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[Two replies - db]

X-Sender: balaz.-at-.prairie.nodak.edu

Date: Fri, 13 Aug 1999 16:10:29 -0500

To: PharmPK.-at-.boomer.org

From: Stefan Balaz

Subject: Re: Integration of compartmental models

Dear Thierry:

Just a comment.

To the best of my knowledge, analytical (explicit, algebraic) integration

of the set of differential equations even for linear models is not feasible

for any structure of the compartment system. Let's denote the set of

underlying differential equations as -dc/dt = B x c + f, where c is the

(column) vector of the drug concentrations in all compartments, B is the

coefficient matrix, f is the (column) vector of the functions characterizing

the drug input to individual compartments, and x indicates multiplication.

Analytical integration can only be done if you can (1) express analytically

the eigenvalues of the matrix B and (2) integrate analytically the functions

"f(t-a) x exp(-b x a) x da" (f are the input functions, b are the

eigenvalues, and a is a variable). There is a lot of literature on the

subject but it is frequently difficult to read for a non-mathematician. A

very good treatise (in German) on the symbolic integration by Bozler,

Heinzel, Koss, and Wolf (Karl Thomae and Boehringer) was published long time

ago in a special issue 4a of Arzneimittel Forschung/Drug Research vol

27 (1977).

Translating the first condition (eigenvalues) into the structure of the

compartmental models, the models with unidirectional transfer are

comparatively easy to integrate analytically but cycles can cause problems.

However, when bidirectional transfer is involved, analytical integration

becomes very difficult, if at all feasible, when your system contains a

series of 3 or more compartments that exhibit bidirectional transfer

(occasionally, serial 4-compartment systems with bidirectional transfers are

integrable but only for special cases when some of the parameters are

identical - an example is given in our paper in J. Theor. Biol. 185 (1997)

213-222). The second condition (integration of the input functions) cannot

be discussed because you do not provide any information about your input.

In summary, numerical integration is the method to go, if you have a

multitude of more complicated models. I hope this helps and saves some time.

Good luck with your simulations,

Stefan

---

X-Sender: walt.-a-.mail.simulations-plus.com

Date: Sat, 14 Aug 1999 15:32:21 -0700

To: PharmPK.-a-.boomer.org

From: Walt Woltosz

Subject: PharmPK Re: Integration of compartmental models

Dear David,

While analytical solutions are fast, they are also far more difficult to

make general. If you want to solve something analytically, then every

dependency in the model must take a functional form that is integrable. It

can be challenging to come up with such models, and very time-consuming if

the problem is complex.

I come from the aerospace industry, which has had the luxury of many, many

millions of dollars and at least three decades of simulation development

for a wide variety of problems. With perhaps a rare exception (at least in

my experience), the most sophisticated models of any phenomenon are those

that involve numerically integrating a set of differential equations. Of

course, sometimes, within the derivative routines, there are closed form

(analytical) models that provide some of the numbers required to generate

the derivatives.

The tremendous flexibility of integrating differential equations allows the

developer to incorporate dependencies that are not always easily put into a

functional form that allows analytical solution (such as table lookups of

dertain dependencies from experimental data). And modifying the model,

which is inevitable, is made much faster as well, because you avoid having

to analytically derive a new solution each time your equations change.

While almost all modeling approaches involve theoretical equations based on

simplifying assumptions, those using the numerical solution approach can

usually be based on less simplification. The price you pay is the time

required to get a better answer. Computer processing power being relatively

cheap nowadays, this is not the problem it once was. I used to run

simulation/optimization programs on the Space Shuttle that took 12-14 hours

(on a Univac 1108 in 1971) which would now run in minutes on my notebook

computer.

For some problems, analytical solutions are appropriate. But as complexity

increases, analytical solutions often provide only a "quick and dirty"

ballpark estimate of behaviors. I believe there is usually a positive

correlation between quickness and dirtiness. On the other hand, anyone

developing a numerical model is remiss if they don't compare it to a good

analytical model now and then as a cross check - the differences should be

explainable.

Walt Woltosz Phone: (661) 723-7723

Chairman & CEO FAX: (661) 723-5524

Simulations Plus, Inc. (OTCBB:SIMU)

1220 West Avenue J

Lancaster, CA 93534-2902

U.S.A.

http://www.simulations-plus.com

walt.-a-.simulations-plus.com - On 16 Aug 1999 at 23:11:10, "Stephen Duffull" (sduffull.-a-.fs1.pa.man.ac.uk) sent the message

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To ask a question...

Walt Woltosz commented:

>For some problems, analytical solutions are appropriate.

But as complexity

>increases, analytical solutions often provide only a "quick

and dirty"

>ballpark estimate of behaviors.

If there is a closed form analytical solution to the set of

ODEs then why would this be "quick and dirty"? Surely the

solution must be just that? If there is such a solution

then (to me) it would seem prudent to use it since this

would eliminate a numerical routine and provide a

significant improvement in speed. Despite comments to the

contrary speed is an important issue for some population

PKPD problems.

Regards

Steve

=====================

Stephen Duffull

School of Pharmacy

University of Manchester

Manchester, M13 9PL, UK

Ph +44 161 275 2355

Fax +44 161 275 2396 - On 18 Nov 1999 at 19:56:57, "Leon Aarons" (laarons.aaa.fs1.pa.man.ac.uk) sent the message

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Several correspondents have commented that it is not necessary to

use analytical solutions to linear ordinary differential equations

in fitting pharmacokinetic models to data as there are several good

programs available that incorporate differential equation solvers.

While this is undoubtedly true and certainly it is an easier way to

implement many pharmacokinetic models, there is a price to pay.

The solution of the differential equation is a numerical procedure

which is implemented within another numerical procedure: the optimizer

within the the nonlinear regression routine. Therefore it will

invariably take much longer specifying the model in terms of

differential equations. For single individual modelling this may not

be a concern but for large mixed-effects (population) problems this

becomes significant. Furthermore for flat or ill-conditioned surfaces

numerical errors arising from the differential equation solver can

propagate leading to nonconvergence and even divergence.

Therefore whenever possible, it is preferable to specify the equation

in algebraic form.

Leon Aarons

_____________

Leon Aarons

School of Pharmacy and Pharmaceutical Sciences

University of Manchester

Manchester, M13 9PL, U.K.

tel +44-161-275-2357

fax +44-161-275-2396

email l.aarons.at.man.ac.uk - On 21 Nov 1999 at 16:14:57, Daniel Combs (dzc.aaa.gene.com) sent the message

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Yes:

Solving ordinary differential equations within a nonlinear regression

program is indeed time consuming even for small datasets. However, the

regression software used may alow direct incorporation of the differential

equation part of the model into a fortran program. This is a feature of

WinNonlin and is said to decrease the run time significantly. It makes

sense, because fortran is very good at doing many complicated nested loops

very quickly. I have not used this feature but it seems a possible

solution to waiting hours for odfe solution and regression convergence.

Perhaps other software such as Adapt and WinSam have the same features.

One draw back, aside from the extra programming, is that one must have a

fortran compliler such as Microsoft Fortran Powerstation 4.0 or Digital

Visual Fortran (version ??).

Other possibilities could be to write the entire model in fortran, or

perhaps use the SAS user application for PK modeling (never officially

released by SAS since it was a user program).

===========

Dan Combs

PK&Metabolism Dept.

Genentech Inc.

voice (650) 225-5847

fax (650) 225-6452

e-mail dzc.aaa.gene.com - On 7 Dec 1999 at 23:09:12, Roger Jelliffe (jelliffe.-at-.usc.edu) sent the message

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Dear Dan Combs:

About the need for differential equation solvers and Fortran

compilers. We

use the BOXES program in the USC*PACK collection. It writes the Fortran

source code for the model part of the analysis. You can then tweak the

source code if you wish. Then you can send this file, and another file with

the data and the instructions, to the Cray T3E at the San Diego

Supercomputer Center. The source code gets compiled. The analysis is done,

either using the iterative Bayesian or the Nonparametric EM software (for

best results, get the assay error pattern determined first, then find

gamma, the remainder of the intraindividual variability, then use this data

in the NPEM program. Results are then downloaded back to your PC and

examined with numbers and plots. This is an NIH supported research

resource, and it avoids having to get a compiler. That is all done for you.

We can arrange an account for those who are interested.

[Please reply directly to Roger if you want an account - db]

Very best regards,

Roger Jelliffe

Roger W. Jelliffe, M.D. Professor of Medicine, USC

USC Laboratory of Applied Pharmacokinetics

2250 Alcazar St, Los Angeles CA 90033, USA

Phone (323)442-1300, fax (323)442-1302, email= jelliffe.at.hsc.usc.edu

Our web site= http://www.usc.edu/hsc/lab_apk

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