Dear BTK software developers:
I'm writing an article on the use of sensitivity anlysis for
experimental design (for the toxicology literature) and am
trying to identify software on which these calculations are
relatively easy. I intend to put a list of the software in
an appendix. I've devised a sample problem to test on various
programs, given below. (Some of the explanation will be
remedial for you, this was written for software companies who
don't know what partial derivatives are.) If you have a software
package for distribution, I would appreciate it if you would try
this problem and provide a demo document or template program for
interested users. (I have a Mac &, with some difficulty can access
a PC, if I must perform the demo's myself. However this will
require becoming familiar with a number of programs and I fear
that I might not represent them properly.) The application should
be fairly easy, and I'm making decisions on what to include on my
list based on ease. (If it can be done, but its a pain in the ***
then I won't list the software.) The target audience is people with
marginal 'programming' skills. Therefore, I will need to see a copy
or description of the programing or model entry/problem definition
dialogue that one must perform to get a solution before I can include
the software in my list. Also, of course, you'll want to provide info
on how to get the software.
By the way, Chemical Engineering Progress (trade journal for the
AIChE) has an annual software directory. If you're interested in
other venues for advertising software, you might contact CEP.
Karen Simpson is Managing Editor, phone: 212/705-7337,
FAX: 212/752-3294 (once you get the USA).
Thanks in advance,
Paul
schlosser at beta.ciit.org
Paul Schlosser
CIIT
P.O. Box 12137
Research Triangle Park, NC 27709
USA
phone: (919) 541-2070, ext. 243
FAX: (919) 541-9015
Simple Example of Sensitivity Analysis for Experiment Design
The example system contains two variables/chemical species
linked by the reaction x1 -> x2, the rate of which is
r = Vmax*x1/(Km + x1).
For illustrative purposes, take Vmax = 2/min and Km = 5 mM.
We will consider initial conditions with x1(0) = x10 and
x2(0) = 0, and incubations up to 90 min in length.
The solution is considered to be a function of the parameter
values; i.e., x1(t; Vmax, Km) and x2(t; Vmax, Km).
In our experiments, concentrations are determined by using
14C labelled compounds, and higher concentrations are obtained
by diluting with cold material, so a measurement of x1 = 1.0 mM
in an experiment with x10 = 10.0 mM is given the same weight
as a measurement of x1 = 0.1 mM in an experiment with x10 = 1.0 mM.
In both cases, the measurement would be that of 10% of the
initial radioactivity (which is the same in both experiments)
being assigned to x1. Therefore, I will define the
measurement variables,
y1(t; Vmax, Km) = x1/x10 and y2(t; Vmax, Km) = x2/x10 .
I want to compute the derivatives,
GKm1 = dy1(t; Vmax, Km)/dKm
and
GKm2 = dy2(t; Vmax, Km)/dKm
for various initial conditions (e.g., for values of x10
between 0.1 and 10.0 mM). These derivatives will be a
function of time. (Alternately, we could consider the
derivatives with respect to Vmax.) I then wish to compute
the Information Index
e(Km, t) = [(w1*GKm1*GKm1) + (w2*GKm2*GKm2)]*Km*Km,
where w1 and w2 are weights (can be taken to w1 = w2 = 1.0
here, but may be different in general). Multiplying by Km2
(i.e., the square of the parameter for which the derivatives
were taken) normalizes results, and would be a common feature.
The object is then to identify the combination of x10 and t
for which e(Km, t) is maximized, and to be able to show a
graph or listing of e(Km, t) as a function of time at that
value of x10. In short, where is the maximum of e(Km, t) in
the region t in [0, 90min] and x10 in [0.1, 10.0mM],
and what does e(Km, t) look like for that value of x10?
(For notational purposes, e(Km, t) may be denote eKm.)
In general, we wish to consider general models in which
pools are linked by fluxes, and in which the fluxes can
be assigned a user-defined function of arbitrary complexity.
The models often contain time-dependent 'control' variables
or parameters, such as might occur under periodic chemical
exposure (e.g., 6 hrs of continual exposure followed by 18 hrs
of non-exposure, or bolus injections every 4 hrs, etc).
