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Hi,
I heard that (especially) for complex PK models, is better to use for
fitting algebraic equations instead of corresponding differential eq.
Is that true? Why?
Thank you,
laurian vlase
Vlase Laurian
Dept. of Pharmaceutical Technology and Biopharmaceutics
Faculty of Pharmacy
University of Medicine and Pharmacy "Iuliu Hatieganu"
13, Emil Isac
Cluj-Napoca, Cluj 3400, Romania
vlaselaur.at.yahoo.com
online on Yahoo Messenger
[Typically the algebraic equations can be solved more quickly than the
differential equations. However you have to know the equations from
somewhere, reference or derivation
(http://www.boomer.org/c/p3/c07/c0707.html). Developing algebraic
equations may require re-parameterization (e.g. k12,k21,k10 to
alpha/beta). With a fast computer and appropriate software (boomer
perhaps - http://www.boomer.org/ or see
http://www.boomer.org/pkin/soft.html for other software) testing
different models may be more efficient using an easily modified
differential equation format - db]
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The following message was posted to: PharmPK
Hi Vlase,
Just adding to David's comments, explicit equations would be your gold
standard, as they give you the exact answer to the problem. However,
many times it is impossible to get the explicit answer to various
equations and that's one of the main reasons differential equations are
used. With modern computer power and availability of various programs
for solving differential equations, it is more practical, and many times
less time-consuming, to set up the model using differential equations
and estimate the model parameters.
Toufigh Gordi
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Thank you all for answer.
I'm asking that because I'm looking for some advantages for using
algebraic equations instead of differential.
Until now, I can say (please correct me if I wrong):
1. Advantafes for using diff:
-as Dr. Gordi says, is more practical and less time consuming
2. Advantages for using algebraic eq:
-for scientist: can examine the time course of a certain compound in a
certain place (compartment) and can see the dependence of concentration
versus all the model constants.
- the fitting, in case of complex models, can be more robust (?),
especially when the data is not reach or have large CV% (?).
I'm not talking about algebraic eq. of some simple models, but about
complex models, when the user probabely will never see the analytic
solution due to complex calculations.
For example, using a simple and fast method of calculation, we have
obtained the algebraic form of all the compound/in every compartment
from the next models:
-PK19, PK42 from PK/PD Data Analysis by Gabrielsson and Weiner
or, for more complex models, like the next one (absorption with
firt-pass effect and sistemic metabolisation):
http://www.geocities.com/vlaselaur/pk/_pk.doc
Please let me know your opinion
--
Vlase Laurian
Dept. of Pharmaceutical Technology and Biopharmaceutics
Faculty of Pharmacy
University of Medicine and Pharmacy "Iuliu Hatieganu"
13, Emil Isac
Cluj-Napoca, Cluj 3400, Romania
vlaselaur.at.yahoo.com
online on Yahoo Messenger
[The first advantage for algebraic isn't correct as this is also
possible with differential equations. With regard to the second
advantage of algebraic equations, differential equation solution will
be slower and numerical approximations, however there are common
situations were the algebraic solution are not robust. For example for
a simple one compartment model with oral absorption you need to include
additional equations as ka = kel. Many (most) algebraic solutions
include the difference between parameters in denominator - db]
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The following message was posted to: PharmPK
Dear Laurian,
With respect to the discussion about 'algebraic equations' (or 'explicit
equations') and differential equations, I do not fully agree with you:
> 2. Advantages for using algebraic eq:
> -for scientist: can examine the time course of a certain compound in a
> certain place (compartment) and can see the dependence of
concentration
> versus all the model constants.
This is a relative advantage; the information can be obtained also from
the
numerical solution of the differential equations.
> - the fitting, in case of complex models, can be more robust (?),
> especially when the data is not reach or have large CV% (?).
I don't think this is fully true. Algebraic solutions provide exact
solutions (depending on implicit numerical accuracy of the software),
which
is indeed an advantage. Numerical solutions are always approximations,
and
this may indeed cause problems, e.g. in the estimation of derivatives
(as
required for many fitting algorithms). This is mainly a technical
question
that usually can be solved adequately by proper design of the software.
If parameters have a large CV%, or if the fitting does not converge (I
trust
that you refer to these situations), the data do not contain sufficient
information about the model parameters, and this has nothing to do with
the
differential equations.
> I'm not talking about algebraic eq. of some simple models, but about
> complex models, when the user probabely will never see the analytic
> solution due to complex calculations.
The main problem with algebraic equations is that you have to derive
(and
test!) the equations for each individual model. Differential equations
describe the problem in relative simple equations, that can be written
easily. In addition, a set of differential equations can be easily
'fed' to
a computer program solving the differential equations numerically. This
allows to fit the parameters of very different and very complex models
with
a single program, by changing only the differential equations.
Please note that I do not see any objection against using algebraic
equations. On the contrary, these are still the gold standard, since the
testing of a program solving numerical equations is eventually done by
comparing the results to that of an algebraic solution. So, if you are
happy
using algebraic solutions, just continue! It is like horses; an
algebraic
solution is like a race-horse; it runs fast and elegant, but requires
much
attention, and needs a proper preparation for each race. A program for
solving differential equations numerically is a work-horse. It runs more
slowly and not really efficient, but it can do any job whatever you
want to
do.
Best regards,
Hans Proost
Johannes H. Proost
Dept. of Pharmacokinetics and Drug Delivery
University Centre for Pharmacy
Antonius Deusinglaan 1
9713 AV Groningen, The Netherlands
tel. 31-50 363 3292
fax 31-50 363 3247
Email: j.h.proost.aaa.rug.nl
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Hi Laurian,
As a general statement and as a rule of thumb I'd like say:
1. In general PK-models are differential equations, or substantial
parts of the model are differential equations
2. In some cases there exist an explicit solution (algebraic form) of
that model. In these cases it is advantageous to do the calculations
using the explicit solution.
3. Using an algebraic equation, that is not the explicit solution of
the differential equation model, is in general disadvantageous, if
compared to the use of the differential equation model itself.
Maybe this is not an answer to your question, but it can help to find
out, what your problem is.
Kind regards
Peter
Peter Wolna
Merck KGaA, Clinical R&D / Clinical Statistics
Frankfurter Str 250
D-64293 Darmstadt
Phone: +49- 61 51- 72 61 68
Fax: +49- 61 51- 72 63 61
Email: peter.wolna.at.merck.de
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The following message was posted to: PharmPK
I agree with Hans and Peter.
Analytical models are fast, but often require simplifying assumptions
and/or splitting the problem into different time segments to handle
discontinuities. Integrating differential equations provides the
greatest flexibility, as a single software package can be developed to
handle a wide variety of problems without the need to rederive
analytical solutions for every new twist.
Walt Woltosz
Chairman & CEO
Simulations Plus, Inc. (AMEX: SLP)
1220 W. Avenue J
Lancaster, CA 93534-2902
U.S.A.
http://www.simulations-plus.com
Phone: (661) 723-7723
FAX: (661) 723-5524
E-mail: walt.-at-.simulations-plus.com
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The following message was posted to: PharmPK
Hi Laurian,
my contribution to this discussion you can find at
http://www.uef.sav.sk/Reply_14_01.pdf
Kind regards,
Maria Durisova, PhD, DSc (Math/Phys)
Vice Director of Institute of Experimental Pharmacology
Slovak Academy of Sciences
and
Head of Department of Pharmacokinetics
Dubravska cesta 9
841 04 Bratislava
Slovak Republic
Tel./Fax: +421 2 54775928
http://www.uef.sav.sk/durisova.htm
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