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Linear and nonlinear system approaches in pharmacokinetics: How much do they have to offer? II. The response mapping operator (RMO) approach

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Abstract

The description of the relationship between different responses measured simultaneously in the same subject is commonly described in terms of specific pharmacokinetic models such as linear compartmental models. An alternative system approach involving response mapping operators (RMOs) is presented. The theoretical and mathematical basis of the RMO approach are derived. The assumptions, limitations, and practical significance of the RMO approach are discussed. The derivation of the RMO is illustrated with several examples. An algorithm and computer program for implementing the RMO in a routine manner is presented. The usage of the computer programs RMO and MAP presented are demonstrated using the pharmacokinetics of trimazosin and cefamandole in humans as examples. The RMO approach offers a new and powerful tool in pharmacokinetic analysis, which allows the investigator to approach certain problems in an objective, rational way without getting involved in specific pharmacokinetic modeling.

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Abbreviations

ai, αi :

Model-independent parameters of response function c1(t)

bi, βi :

Model-independent parameters of response function c2(t)

c 1(t),c 2(t):

Response functions

c δ1 (t),c 2(t):

Unit impulse responses corresponding to c1(t) and c2(t)

c ix(t):

Subresponse function

D iv :

Intravenous bolus dose

â i ,\(\hat \alpha _i\) :

Model independent parameter of response functionĉ 1(t)

f(t):

Input rate/precursor function

gi, γi :

Parameters calculate from c1(t) response function for cases wherer >0.

K 1,K 2 :

“Derivative numbers” for the c1(t) and c2(t) response functions

L :

Laplace transform operator

L{ }:

Linear operator

L :

=n 2+r

m i :

=K i+3, iε 1,2

n1, n2 :

Number of exponential terms inc 1(t) andc 2(t)

P(x):

Polynomial

Φ(t):

Auxiliary function

r :

=n 1-m 1+1=n 1-K 1-2

R(x):

Polynomial

r 0 :

Constant infusion rate

s:

Laplace transform variable

t :

Time

τ :

Time for infusion stop

u i,v i :

Parameters of the mapping function

w(t):

Mapping functionw(t) ≡L −1{L{c 2(s m L{c1})}

x(t):

Arbitrary function common to the responses considered

i,j, k, r :

Regular array subscripts

δ,f :

Subscripts denoting an impulse (δ) or nonimpulse (f) input

^:

Is used to denote “predictor responses”, (and associated parameters) i.e., responses which are not used as reference responses to determine the RMO

(i):

ith derivative with respect to time

*:

Convolution operation [x(t)*y(t)≡∫ t0 x(t−u)y(u)du]

E i :

Polynomial operator

L{ },L −1 :

Laplace and inverse Laplace transform operators

RMO 1,2{ }:

Response mapping operator (ĉ 2(t)=RMO 1,2{ĉ 1(t)})

Π:

Continued product operator

∑:

Summation operator

References

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Author information

Authors and Affiliations

  1. College of Pharmacy, University of Iowa, 52242, Iowa City, Iowa

    Peter Veng-Pedersen

  2. Department of Pharmacy, School of Pharmacy, University of California, 94143-0446, San Francisco, California

    Ronald A. Siegel

  3. Merrell Dow Research Institute, 2110 E. Galbraith Road, P.O. Box 156300, 45215-6300, Cincinnati, Ohio

    Harold Boxenbaum

  4. The Upjohn Company, 49001, Kalamazoo, Michigan

    Carl M. Metzler

Authors
  1. Peter Veng-Pedersen
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  2. Ronald A. Siegel
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  3. Harold Boxenbaum
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  4. Carl M. Metzler
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Additional information

Part I of this article appeared inJ. Pharmacokin. Biopharm. 16(4); commentaries on both parts and a rebuttal by P.V.P. will appear in JPB, Vol. 16, No. 6.

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Veng-Pedersen, P., Siegel, R.A., Boxenbaum, H. et al. Linear and nonlinear system approaches in pharmacokinetics: How much do they have to offer? II. The response mapping operator (RMO) approach. Journal of Pharmacokinetics and Biopharmaceutics 16, 543–571 (1988). https://doi.org/10.1007/BF01062384

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