- On 27 Apr 1999 at 13:20:59, "Huang, Bill " (Bill.Huang.aaa.MKG.com) sent the message

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All,

Anyone can tell me where I can download free software(s) to

perform convolution analysis. Thank much.

Bill - On 29 Apr 1999 at 22:23:05, ml11439.-at-.goodnet.com (Michael J. Leibold) sent the message

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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] - On 1 May 1999 at 19:09:29, "Joseph Balthasar" (jbalthasar.aaa.pharm.utah.edu) sent the message

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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

****************** - On 4 May 1999 at 21:21:53, Maria Durisova (exfamadu.aaa.savba.sk) sent the message

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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 - On 5 May 1999 at 22:32:53, ml11439.aaa.goodnet.com (Michael J. Leibold) sent the message

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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 - On 7 May 1999 at 11:52:02, "L. Dedik" (DEDIK.aaa.kam1.vm.stuba.sk) sent the message

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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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