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All,
Anyone can tell me where I can download free software(s) to
perform convolution analysis. Thank much.
Bill
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Hello Bill,
Convolutional analysis is involved in Laplace transform solution of
differential equations. It is particularly applicable to the first order
linear differential equations implemented in pharmacokinetics. Software may
have to obtained through mathematical or engineering channels, since most
pharmacokientic software does not involve Laplace or integral solutiion of
differential equations.
The USC Pack can solve differential equations in terms of compartmental
models. Solution of compartmental models in terms of differential equations
is also a form of convolutional analysis. However, the USC pack is free in
demo form only. The demo form can be obtained on the same web site as
the Boomer package; [ http://www.boomer.org/pkin/ ]
Mike Leibold, PharmD, RPh
ML11439.-at-.goonet.com
[The direct USC Pack URL is
http://www.usc.edu/hsc/lab_apk/software/uscpack.html which is listed from
the http://www.boomer.org/pkin/ address - db]
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Bill:
If you are interested in convolution / deconvolution in the Laplace domain,
you may wish to look into the "Scientist" program by MicroMath. The
Scientist allows convolution in the Laplace domain to simplify the task of
building Laplace transforms (e.g., L(C)= in(s) * d(s) [where in(s) and
d(s) are Laplace transforms of the input and disposition functions,
respectively]). The Scientist's manual is directed to a broad audience
(chemists, engineers, etc.); however, it is quite easy to use and the
manual often provides examples of PK applications.
I also believe that Peter Veng-Pedersen may have some software available
for convolution / deconvolution which he has applied to his disposition
decomposition analyses.
Hope this helps. Good luck.
J. Balthasar
******************
Joseph P. Balthasar, PhD
Assistant Professor
University of Utah
Department of Pharmaceutics
and Pharmaceutical Chemistry
421 Wakara Way, Suite 316
Salt Lake City, Utah 84108
Telephone: 801-585-5958
Fax: 801-585-3614
******************
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>Convolutional analysis is involved in Laplace transform
>solution of differential equations. It is particularly
>applicable to the first order linear differential equations
>implemented in pharmacokinetics.
In principle, the fact that the n-order (where n is a finite
integer number >0) differential equations can be solved using
the Laplace transformation, has nothing to do with the
convolution.
The term convolution comes from the Latin expression "cum
volvere", meaning to intertwine.
For the two discrete functions:
X[k], i.e. (X[1], X[2],....X[k],...X[N])
and
H[k], i.e. (H[1], H[2],....H[k],...H[N]),
the convolution Y[k] is given by
Y[k] = summation from i=1 to k of X[i].H[n-(k-i)]. (1)
For the two continuous functions X(t) and H(t) the convolution
Y(t) is given by integral equation
Y(t)=integral from 0 to t of X(tau)H(t-tau)dtau. (2)
This equation is often denoted using the notation
Y(t)=X(t)*H(t). (3)
Integrals of the type given by Eq. 2 are known in the
transform theory as the convolution integrals and in the
classical mathematics as the Duhamel integrals (J. M. C.
Duhamel (1797-1872) was a French mathematician whose most
important work was on partial differential equations.)
If the functions X(t) and H(t) are the Laplace transformable
functions, and X(s) and H(s) are the Laplace transforms of
these functions, Eq.2 can be written in the Laplace domain in
the form of Eq. 3
Y(s)=X(s).H(s),
where Y(s) is the Laplace transform of Y(t).
Best regards,
Maria Durisova
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Bonne Jour Maria,
The convolution theorem states that if f(t) and g(t) are the
inverse transforms of F(s) and G(s), respectively, and that the
inverse of the product H(s)= F(s)G(s) is the convolution of f(t)
and g(t) written (f*g)(t).
Multiplying the Laplace transforms of the input In(s) and
disposition functions d(s) involved in pharmacokinetics is followed
by taking the inverse transform of the product: in(s)d(s). This
is the convolution of in(t) and d(t) as described the above
convolution theorem. The proof of this theorem shows that the
product of the two Laplace transforms is equal to the laplace
transform of the convolution itegral.
System analysis considers the input and output of a linear
system to be related by the convolution integral of the input
and output functions. The convolution integral weighs the past
values of the input to the present value of the output.
However, the convolution integral is difficult to evaluate
analytically except for simple functions, and thus, use is made
of the corresponding Laplace transform of the convolution integral.
Since we know that the product of the Laplace transforms of the
input and disposition functions is equal to the Laplace transform
of the convolution itegral, the inverse transform is equal to the
solution of the convolution integral.
Convolution integral:
Y(t)=integral from 0 to t of X(tau)H(t-tau)dtau.
In summary, all integrated linear pharmacokinetic equations are
solutions of the convolution integral of the input and dispostion
functions. Laplace transforms make the various complex solutions
possible. According to one paper, analysis of similar problems
with the convolution integral itself, could be called convolutional
analysis.
Mike Leibold, PharmD, RPh
ML11439.aaa.goodnet.com
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I do not want needlessly to prolong this discussion, however
the two following comments may be useful:
> Convolutional analysis is involved in Laplace transform
> solution of differential equations. It is particularly
> applicable to the first order linear differential
> equations...
1.
The convolution analysis is a time domain procedure, while
the Laplace transform transforms functions with a real argument
(e.g. ordinary differential equations) to functions with
a complex argument.
The Laplace transformation can be used to simplify the
solution of the n-order differential equation and/or of the
convolution integral. If it is used to solve the n-order
differential equation, the time domain differential equation
is transformed into the algebraic equation in the s domain
(where s is the Laplace variable). After the solution of this
algebraic equation in the s domain, the solution of the
n-order differential equation in the time domain can be
obtained employing the inverse Laplace transformation.
Analogously, the Laplace transformation can be used to solve
the convolution integral, since this solution
may be difficult in the time domain, as you pointed out:
> However, the convolution integral is difficult to
evaluate...
2.
If the function H(t) in Eq. 1 is the weighting
function of the n-order model of the system describing the
drug fate or effect after the drug input X(t), the output
Y(t) of this n-order model can be obtained according to this
equation
Y(t)=integral from 0 to t of X(tau)H(t-tau)dtau. (1)
In bio-medicine very frequently an inverse procedure is used,
to obtain the weighting function. Our study (Dedik, L.,
Durisova, M., Comput. Meth. Programs Biomed., 51, 1996,
183-192) describes the procedure for the determination of
weighting functions in the analytical form of various simple
and/or composite models (up to ninth-order models). This
procedure is applicable even in situations in which the
deconvolution methods cannot be used.
Sincerely,
Maria Durisova
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