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Dynamics of drug distribution. I. Role of the second and third curve moments

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Abstract

Conventionally, the dynamics of distribution in the body is evaluated by the so-called distribution half-life (e.g., t 1/2,α but then the mean time of the distribution process is underestimated due tothe influence of elimination. By contrast, information about the dynamics of distribution contained in drug disposition curves can be extracted by the second and third curve moments, parameters that are related to the variance (VDRT)and skewness (SDRT)of residence time distributions; whereas the equilibrium state characterized by the volume of distribution (Vss), isdetermined by the mean residence time (MDRT)or the first curve moment. The approach represents a general noncompartmental analysis that is independent of a detailed structural model or a particular disposition function. Two parameters are introduced to characterize the dynamics of drug distribution: (i)the degree of departure of the system from “well-mixed” behavior of instantaneous distribution equilibrium (related to VDRT)and (ii)the mean time until equilibration is achieved (mean equilibration time, MEQT),which additionally depends on SDRT.Both parameters are quantitative measures of the dynamics of distribution and display explicit physical significance in terms of distribution within the corresponding noneliminating system. It is further shown that the so-called “distribution phase” in biexponential disposition curves is related to a monoexponential mixing curve of its corresponding noneliminating system with an equilibration or mixing half-time, t 1/2,M =t 1/2,α (Vβ/V * ss ), where V * ss denotes the distribution volume of the noneliminating system. The results are applied to mixing and disposition curves measured for acetaminophen in liver-ligated and intact rats, respectively.

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Abbreviations

A e :

Amount eliminated

AUC :

Area under the curve

\(\widehat{AUC}\) :

ObservedAUC

AUMC :

First moment of disposition curve (=MO D1

C(t) :

Blood (plasma) concentration

C D(t):

Disposition curve (eliminating system)

C M(t):

Mixing curve [C *(t)-C*(∞), noneliminating system]

C C(t):

Circulatory single-pass curve

C *(t):

Distribution curve approaching C*(∞)

CL :

Clearance

CS 3 :

Relative skewness

CTD :

Circulation time distribution

CV 2 :

Relative dispersion

D :

Dose (bolus intravenous injection)

°(t) :

Impulse function (Dirac's delta function)

θ :

Weighting parameter for biexponential distribution

E :

Body extraction ratio

Ex[T] :

Expected value of the random variableT

F(t) :

Cumulative distribution

f=dF/dt :

Density function

f(s) :

Laplace transform off(t)

λ :

Exponential coefficient

m(t) :

Renewal density

M(t) :

Renewal function

MCT :

Mean circulation time

MDRT :

Mean disposition residence time

MEQT :

Mean equilibration time

MO k :

Moment ofkth order

N(t) :

Number of recirculations upon timet

N ss(t):

Number of recirculations at steady state

N e :

Total number of recirculations (until elimination)

P :

Probability

Q :

Cardiac output

RTD :

Residence time distribution

S :

Pharmacokinetic system (eliminating orRTD system)

S * :

Noneliminating system

S c :

Eliminating circulatory single-pass (CTD) system

S *C :

Noneliminating circulatory single-pass (CTD) system

S k :

Random residence time until thekth recirculation

SCT :

Skewness of circulation time distribution

SDRT :

Skewness of disposition residence time distribution

T :

Random residence or transit time of a molecule

T c :

Random circulatory transit time of a molecule

t ci :

Circulation time of the ith circulation

T D :

Random disposition residence time

t 1/2,α :

Half-life of the alpha phase [C D (t) biexponential)]

t 1/2,M :

Distribution half-life [CM(t) monoexponential]

Vβ :

Terminal distribution volume (β-phase)

V ss :

Steady state distribution volume

VCT :

Variance of circulation time distribution

VDRT:

Variance of disposition residence time distribution

*:

Noneliminating system

A:

Arterial sampling

C:

Circulatory single-pass system

D:

Disposition system (recirculatory)

M:

Mixing curve (noneliminating system)

V:

Peripheral venous sampling

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Authors and Affiliations

  1. Faculty of Pharmacy, University of Toronto, Toronto, Canada

    Michael Weiss & K. Sandy Pang

Authors
  1. Michael Weiss
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  2. K. Sandy Pang
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Additional information

This work was supported by the Medical Research Council of Canada. M. Weiss was a Visiting Rosenstadt Professor at the University of Toronto. K. S. Pang is a recipient of the Faculty Development Award, Medical Research Council of Canada.

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Weiss, M., Pang, K.S. Dynamics of drug distribution. I. Role of the second and third curve moments. Journal of Pharmacokinetics and Biopharmaceutics 20, 253–278 (1992). https://doi.org/10.1007/BF01062527

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