BIOPHARMACEUTICS & DRUG DISPOSITION, VOL. 12, 1-15 (1991)
REVIEW ARTICLE
REGIONAL PHARMACOKINETICS
111. MODELLING METHODS
RICHARD N . UPTON,*? WILLIAM B . R U N C I M A N AND
~
LAURENCE E. MATHER$
?Department of Anaesthesia and Intensive Care, University of Adelaide, Adelaide 5001, Australia
$Department of Anaesthesia and Intensive Care, Flinders University of South Australia,
Bedford Park 5042, Australia
ABSTRACT
Regional pharmacokinetics is the study of drug concentrations in specific regions of
the body due to drug uptake and elution. Mathematical methods of interpreting regional
pharmacokinetic data can vary greatly in their complexity depending o n their intended
use (i.e. to describe or predict), but must reinforce rather than replace experimental
pharmacokinetics. ‘Black box’ analysis provides and empirical method for the study
of complex pharmacokinetic systems using either statistical moment or linear systems
analysis. However, these methods are only applicable to linear and time-invariant
systems, and ignore the large body of information concerning the physiological and
physicochemical basis of regional pharmacokinetics. Clearance concepts are suitable
for describing linear drug uptake processes, but mass balance principles have wider
applications in describing the rate and extent of both drug uptake and elution. Compartmental models of a region can vary from single compartment descriptions based on
the concept of venous equilibrium to complex multi-compartmental models of the intravascular, interstitial, and intracellular spaces, in which drug transport between coinpartments is a function of drug binding and ionization. Ultimately, as more regional
pharmacokinetic information is obtained, more complex three dimensional models may
be necessary such as those used to describe the uptake of oxygen from capillaries.
KEY WORDS
Pharmacokinetics Drug uptake Tissue distribution Mathematical
methods Modelling
INTRODUCTION
In a preceding paper,’ regional pharmacokinetics was defined as the study
of the factors influencing the time-course of drug concentrations in specific
regions (e.g. tissues, organs) of the body. The drug concentrations in such
regions are a function of the uptake of the drug from the blood entering the
region, and the elution of the drug into the blood leaving the region. Regional
pharmacokinetics allows greater insight into the mechanisms of drug disposition
in the body, and how disease processes affect these mechanisms, than systemic
* Addressee for correspondence:Dr R. N. Upton, Department of Anaesthesia and Intensive Care.
Royal Adelaide Hospital, University of Adelaide, North Terrace, Adelaide 5001, Australia.
0 142-2782/9 1/O10001-1 5$07.50
0 1991 by John Wiley & Sons, Ltd.
Received 25 May 1990
2
R. N . UPTON, W. B . RUNCIMAN AND L. E. MATHER
pharmacokinetics. The physiological and physicochemical basis of regional
pharmacokinetics, and the experimental methods that may be used in the study
of regional pharmacokinetics have been reviewed.'.' The aim of this review
is to provide an overview of pharmacokinetic modelling, and then summarize
regional pharmacokinetic modelling methods so that their advantages and disadvantages can be viewed in perspective.
AN INTRODUCTION TO MATHEMATICAL PHARMACOKINETIC
MODELS
Pharmacokinetic models are simplified representations of biological systems
in which there are time-dependent changes in drug concentration. An essential
feature of all models is that they are simplified by certain assumptions about
the system. There is a large volume of literature concerning pharmacokinetic
modelling, to the extent where some books deal almost entirely with the
properties of models3 In contrast, some authors are blatantly hostile towards
the concept of pharmacokinetic modelling, perhaps feeling that it has overshadowed the experimental basis of pharmacokinetic~.~Less extreme
approaches can be realized by considering a pharmacokinetic model of an
experimental system to be analogous to a map of a city. The map is useful
for planning a route, and helps give an overview of the area, but the map
must be constantly updated as streets change, and any discrepancy between
the streetscape and the map is, of course, attributed to the map. Himmelblau
and Bischoff, in their text book on chemical engineerir~g,~
advocated a pragmatic
approach to modelling in general which is equally applicable to pharmacokinetics: 'We must be wary of attaching to a model a general aura of validity
which it does not merit. It may become acceptable as dogma with no real
basis; this is a disease of mathematical modelling. (These) dangers must be
circumvented by constantly using common-sense in the interpretation of
mathematical results.'
SOME PROPERTIES OF MATHEMATICAL PHARMACOKINETIC
MODELS
All mathematical models consist of variables, parameters, and expressions.
The commonly used pharmacokinetic models are mathematical expressions
that describe the relationship between time (the independent variable) and drug
concentration (the dependent variable). Parameters are other factors included
in the expression that influence the relationships between the variables. It is
of value to briefly discuss the properties of mathematical pharmacokinetic
models.
REGIONAL PHARMACOKINETICS
3
Linearity
Linearity is an attribute of pharmacokinetic models in which rates of drug
transport are proportional to drug concentrations. Biological processes can
be linear (e.g. diffusion)6 or non-linear (e.g. saturable binding and MichaelisMenten enzyme kinetics).' Linear pharmacokinetic models are easier to solve
mathematically than non-linear models, and are therefore preferred unless there
is evidence of non-linearity. Data should be examined for linearity and therefore
suitability for linear modelling by using the principle of superposition.8
Statistical reality
The statistical reality of a model is the attribute that describes how the model
accommodates the randomness of biological processes. Stochastic models
recognize that a single drug molecule has intrinsically uncertain movements
in the body and it is only possible to describe the behaviour of a population
of drug molecules using statistical or probability terminology. In contrast, in
deterministic models, each parameter is assigned definite values and no probability terms are used. While it is generally accepted that all biological systems
are stochastic in nature, at least at a microscopic level,9 deterministic models
are usually adequate in situations where the variability of a parameter is relatively small.
Physiological reality
The physiological reality of a pharmacokinetic model is a measure of how
closely the structure of the model resembles the biological system it represents.
Models which lack physiological reality are generally simple and have a small
number of parameters which generally do not correspond to any particular
biological feature. The physiological reality of a model is increased by introducing more parameters which represent particular biological features (e.g.
organ volumes, blood flows, etc.) and by basing the expressions relating the
parameters on biological principles. This is associated with an increase in the
complexity of the model, both in the construction and use of the model and
in the collection of data to test the model. However, physiologically realistic
models can be used to predict the effect of pathophysiological perturbations'"
and can be used to scale drug kinetics between species."
THE USES OF PHARMACOKINETIC MODELS
Description
Although it is possible to describe and compare drug concentration-time
data graphically, it is often preferable to choose a model that represents the
data, and then compare the simpler features of the model. In this way, the
4
R. N . UPTON, W . B . RUNCIMAN AND L. E. MATHER
data may be reduced to one or two numbers (e.g. mean transit time, extraction
ratio). The ability of a model to describe data is often decided by non-linear
least squares regression analysis.12.13However, the best statistical fit of a model
to data by this method does not guarantee the choice of the most appropriate
model. l 4
Predict ion
If a region is adequately represented by a pharmacokinetic model, the model
may be further used for the prediction of drug concentrations between sample
times or at times outside the experimental period, or it may be used to predict
the drug concentrations expected if some aspect of the system has changed.
This is of use if the model is physiologically realistic and changes in the parameters of the model represent definable changes in the corresponding region
in vivo.
REGIONAL PHARMACOKINETIC MODELS
Model independent methods
A mathematical manipulation of data is often described as being either ‘model
dependent’ or ‘model independent’. The meaning of these terms have been
discussed by Rescigno and Beck.I4 They defined the ‘modulation’ of data as
the modification of the data in accordance with certain criteria which may
involve a change in the information content of the data. Simple examples of
modulation include ruling straight lines between the data points on a graph,
or assuming that data are normally distributed. Clearly, even these simple procedures involve making a hypothesis about the data, and therefore apply a
model to the data in some way. For this reason ‘black box’ and other methods
sometimes called ‘model independent’ are included in this section on modelling.
‘Black box’ methods
Many of these methods have been adapted from methods used in tracer
kinetic analysis,15which in turn were adapted from methods used in chemical
engineering.16Black box analysis provides an empirical and stochastic method
for the study of complex systems. In regional pharmacokinetic terms, this
requires assuming that while the input to a region (the time-course of arterial
concentrations) and the output from a region (the time-course of regional
venous concentrations) can be measured, the factors affecting the nature of
the drug transport and uptake and elution within the region are unknown.
Thus, the region is treated as a ‘black box’ which cannot be ‘opened’. By
their very definition, these methods disregard the large body of information
about the physiological and physicochemical basis of regional pharmacokinetics
reviewed in an earlier paper in this series.I
REGIONAL PHARMACOKINETICS
5
Tracer kinetic principles. The basis of these methods is best demonstrated
by example. Consider an experiment in which a bolus of drug is injected instantaneously into the arterial blood vessels supplying a region which has a nonrecirculating blood supply. A population of drug molecules will enter the region
simultaneously, but due to differences in the vascular paths and the extent
of passage into extravascular tissue of individual molecules, there will be a
range of times taken for individual molecules to emerge from the region and
appear in the venous blood. The distribution of probabilities of a drug molecule
appearing in the venous blood after a given time interval (i.e. with a given
transit time) has been defined as the transport function (h(J of the drug for
the region.I6 If a drug only distributes into a region, the area under the transport
function of the drug is 1.0 (that is if sufficient time is allowed, all drug molecules
will eventually leave the region). The transport function is related to the venous
concentration-time curve for the region by the following equation:
) the venous concentration at time t , Dose is the mass of the drug
where C ( f is
injected, and Q is the blood flow through the region. Thus, the time-course
of C(l)is the same shape as /I(,,, but is scaled for the dose and is inversely
related to blood flow. Rearrangement of this equation for Q gives the classical
indicator dilution method for determining blood flow.
There are several common manipulations of the transport function. Firstly,
the integral of the transport function ( H ( f )is) the distribution of the cumulative
residence times, and represents the probability of a single molecule remaining
in the region for time t or less. Thus, H ( f )is the fractional amount of the
dose that would be collected in a bucket at the venous outlet of the region
or H*,,,) represents the probability
by time t. The residue function (1 of a molecule being in the region at time t . This is the curve recorded by
external counting of an organ after tracer administration to determine blood
flow,” and is an index of the tissue concentrations of the drug. Finally, the
emergence function (q(J is the rate of elution of the drug from the region
at time t and can be calculated from the following equation:
These tracer kinetic principles do not in themselves have many regional pharmacokinetic applications in vivo, except perhaps for the interpretation of nonrecirculating studies of artificially perfused regions. However, they are the basis
of a number of common regional pharmacokinetic methods.
Statistical moment analysis. The shape of the transport function reflects the
interaction of the drug with the region: a drug which is reversibly bound in
6
R. N . UPTON, W. B. RUNCIMAN AND L. E. MATHER
a region will have a broader and flatter transport function than a drug which
cannot leave the capillaries of the region. The shape of the transport function
can be characterized in the same way as most frequency distributions. The
central tendency of the function is given by the mean, in this case the mean
transit time, while the dispersion of the distribution can be characterized by
the standard deviation, or the variance of the transit times. Using standard
statistical methods,’* these can be derived from the moments of the frequency
distribution which are defined as follows: the zero moment = C(!),the first
moment = C(!).t and the second moment = C(().tZ. The higher statistical
moments are not usually calculated for pharmacokinetic data because they
are too variable when the data are raised to powers greater than 2. The moments
can be used to describe the transport function using the following equations:
m
AUC=I
C(,,.dt
(3)
0
MTT=~
0
t . C,,,.dt
(4)
~
AUC
VTT =
Where C(!)is the drug concentration at time f, AUC is the total area under
the concentration-time curve, MTT is the mean transit time, and VTT is the
variance of the transit times.
The mean transit time through a region is related to the blood flow through
the region (Q) and the volume within the region in which the drug apparently
distributes (V):
MTT=-
V
Q
V is not the total volume of the region unless the drug is known to distribute
rapidly and homogeneously throughout the entire region but, for certain drugs,
it can correspond to particular physiological volumes such as the interstitial
volume.
At this point, the discussion of the application of statistical moment analysis
to pharmacokinetics becomes complicated by the wide range of applications
of the analysis. The preceding discussion applies to a region with afferent and
efferent blood vessels and with no recirculation. Because it is possible to apply
statistical moment analysis to almost any pharmacokinetic system, it is essential
to specify the entrance and exit points of the drug molecules, and the system
through which they transverse. It must be made clear whether the system is
recirculating19 or non-recirculating, and whether the measured drug concentration is the eflerent drug concentration from the system (equivalent to A(,)
REGIONAL PHARMACOKINETICS
7
above), or is the drug concentration within the system (equivalent to H*(,)
above). Failure to do so creates ambiguity and ultimately confusion.20
By treating the time-course of systemic drug concentrations after a bolus
as a statistical distribution of the ‘residence’ times of drug in the body, some
workers have used statistical moment analysis to calculate the mean ‘residence’
time of drug in the body.21It must be stressed that this mean residence time
is distinctly different to the concept of residence discussed previously and described by the function H(,).Furthermore, it must be emphasized that statistical
moment analysis is only applicable to systems in which the drug administration
was essentially instantaneous (e.g. rapid bolus injection, or single dose oral
absorption). If this is not the case, deconvolution must be used to obtain
interpretable statistical moment data relating only to drug disposition.
In contrast to its application to systemic p h a r m a c ~ k i n e t i c s , ~
statistical
~,~~
moment analysis based on the tracer kinetic principles discussed previously
has recently been applied to regional pharmacokinetics. The statistical moment
analysis of the mitomycin concentrations in perfusate emerging from an isolated
perfused muscle preparation, after a rapid injection into the afferent p e r f ~ s a t e , ~ ~
gave a mean transit time of between 1 and 2min. McNamara et aLz5presented
a similar method for calculating the MTT of a drug using regional tissue drug
concentrations, and Hori et a1.26used statistical moment analysis to analyse
the kinetics ofp-aminohippurate in the isolated perfused rat kidney.
Although these regional pharmacokinetic uses of statistical moment analysis
are useful for describing an uptake and elution process using one or two
numbers, they are not as informative as more physiologically realistic methods
which can test hypotheses about the mechanisms of drug uptake and elution.
In some cases, however, the parameters of statistical moment analysis can be
compared to the parameters of specific pharmacokinetic models,27in contrast
to their use in black box analysis.
Linear systems analysis-convolution and deconvolution. Equation (1) provides a simple method for predicting the venous concentration-time curve for
a region for which the transport function is known, provided drug enters the
region as an instantaneous bolus. However, in practice it is often difficult to
deliver a drug as an instantaneous bolus, for practical reasons and because
of dispersion at the injection site.I6When administration is not instantaneous
and the afferent arterial concentration-time curve can be measured, the
mathematical technique of convolution can be used to predict the venous concentration-time curve from the arterial concentration-time curve and the transport function, provided that the system is linear (i.e. the principle of
superposition therefore applies) and no aspects affecting drug disposition
change with time (i.e. the principle of time invariance applies). In view of
the large body of literature describing pharmacological systems that are either
non-linear and/or time-invariant, the onus is upon each pharmacokineticist
to show that these requirements are met for the system under study.
8
R. N . UPTON, W. B . RUNCIMAN AND L. E. MATHER
Convolution is described by the following equation:
Cm
where Cv is the venous concentration, F is the transport function, Ca is the
arterial concentration, t is time and T is an arbitrary time interval of integration.
The smaller the value of T, the greater the accuracy of the convolution. This
process can be considered quite simply graphically, and the derivation of this
equation from first principles is contained in Shipley and Clark.2sConvolution
can be done algebraically if the equations describing both the input function
and the transport function are known. However, in most pharmacokinetic
situations only experimental concentration-time points are available and it
is necessary to perform the convolution by numerical methods.
The process of deconvolution can be used to determine the transport function
(F)from the arterial and regional venous drug concentrations, a common experimental system in regional pharmacokinetics. Several methods are available
for estimating the transport function, but all require that the drug kinetics
in the region are linear and time-invariant. The simplest of these is by iteration
whereby the transport function is assumed to be described by a particular
type of equation. A computer is used to successively modify the parameters
of the equation until the best fit to the output function is obtained. A lagged
normal density curve, random walk equation, log normal curve, polynomials,
and gamma variate have all be used to describe transport function^.'^.^^ The
pitfalls of these methods when the data are ‘noisy’ have been d o c ~ m e n t e d . ~ ~
A more complex approach to deconvolution is the use of transforms of the
input (arterial) and output (venous) functions into another domain. Laplace,
Z, and Fourier transformations all have been used.”
Alternatively, the moment generating function16 states that the moments of
the output function is the sum of the moments of the input function and the
transport functions. Thus, the MTT and VTT, and therefore the approximate
shape of the transport function, can be calculated from the following equations:
MTT,,, MTT,,,,, MTT,,;, are the mean transit times of the transport, output,
and input functions, respectively, and VTT represents the variance of the transit
times for the corresponding functions.
Deconvolution and statistical moment analysis can be used on the same
data sets (in each case to determine some feature of the transport function),
but deconvolution has a number of advantages. Firstly, it is not necessary
to deliver the drug as a rapid bolus directly into the afferent blood supply
of a region in a non-circulating system, but rather it is only necessary to characterize the time-course of the arterial and regional venous drug concentrations-a
REGIONAL PHARMACOKINETICS
9
more common and less complicated experimental method. Secondly, the entire
shape of the transport function is described, in contrast to statistical moment
analysis which describes only the mean and variance of the transport function.
Convolution and deconvolution may be a useful method for describing or predicting the regional kinetics of a drug in a region which is exposed to different
time-courses of arterial concentrations. This is done by deconvoluting the
experimental data to determine the transport function, and subsequently convoluting the transport function with the new arterial function. This could be
an important, empirical method for ‘modelling’ particular regions in physiological pharmacokinetic models32which are not of specific interest in a particular
study, but whose contribution to the arterial and other venous blood concentrations of the body must be known.
Clearance conceprs. Clearance concepts33 were developed as an alternative
to using half-lives and rate constants to describe drug elimination. Drug clearance is treated as the rate of drug elimination normalized for the drug concentration entering the region, which, in regional pharmacokinetic terms, is
measured by the product of the regional blood flow and drug extraction ratio.34
Clearance concepts are therefore a relatively physiological and regional description of drug elimination and have been used to investigate the effects of enzyme
activity and protein binding on hepatic elimination. However, the terms clearance and extraction ratio are not suitable for describing systems which are
non-linear, are not at arterial steady-state drug concentration, or in which
regional drug uptake is not due to elimination.
Muss balance principles. A limitation of clearance concepts is that clearance
is only suitable for describing linear systems. However, this limitation can be
avoided if the Law of Conservation of Matter used to derive clearance concepts
is applied directly to regions of the body.35 Two fundamental terms can be
derived which can describe both drug uptake and elution, whether it is due
to either drug elimination or distribution. This is done by considering a hypothetical region with a regional blood flow of Q. If drug enters this region
at a concentration C, in the arterial blood, and leaves the region at a concentration C, in the regional venous blood, the fluxes of drug (i.e. mass per unit
time) in the vessels entering and leaving the region are C, . Q, and C,.Q, respecis therefore:
t i ~ e l y . ~The
~ . ~net
’ drug flux (Jnel)
This will be positive when there is net movement of drug from the blood into
the extravascular space of a region (drug uptake) and will be negative when
there is net movement out of the region into the blood (drug elution). A plot
of the time-course of Jne,is a ‘net flux plot’.
The extent of uptake can be approximated by integrating J,,, with respect
to time, starting typically from the time before the drug dose when the blood
10
R. N . UPTON, W . B . RUNCIMAN AND L. E. MATHER
drug concentrations were zero. This is defined as the net mass of drug that
has entered a region via the arterial vessels, but has not left the region via
the venous blood vessels (Mnet).Integrating equation (10) over the time interval
0 to t gives the value of M,,, at time t :
A plot of the time-course of M,,, is a ‘net mass plot’, and the slope of this
plot is the net drug flux (Jnet).
If a number of criteria are to be satisfied, Jnetwill approximate the rate,
and M,,, the extent, of drug uptake and elution. Firstly, the contribution of
the transit of blood through a region to J,,, must be negligible. Except in
studies of very rapid drug uptake (i.e. occurring over a time-course of seconds),
this contribution is negligible. Secondly, all drug other than that undergoing
elimination must enter and leave the region via the blood vessels. It has been
shown that the rate of diffusion of drugs from the surface of organs’* and
the rate of removal via lymphatic vessels39is negligible compared to that by
vascular transport. Finally, the afferent and efferent blood samples must be
representative of blood respectively entering and leaving the region, and the
time-course of the arterial and venous blood concentrations must be adequately
defined.
The usefulness of these mass balance terms in describing regional pharmacokinetics has recently been d e m o n ~ t r a t e d and
, ~ ~ indeed it is possible to show
that the ‘clearance concept’ of regional drug clearance is the regional drug
flux normalized for the arterial drug concentration. The principles of mass
balance were used by Kety and Schmidt4”to determine brain blood flow and
are extensively discussed in the tracer kinetic literat~re.”.~’
However, the use
of mass balance principles in regional pharmacokinetics has been relatively
uncommon. A study of the uptake of thiopentone by the human brain was
reported by Price et u I . , and
~ ~ the uptake of pethidine into the brain of fetal
The review by Horowitz and Powell‘’‘’
lambs was examined by Szeto et
cites several examples, based on the study of myocardial drug uptake by Selden
and Nei11.45The use of mass balance principles in a chronically catheterized
sheep preparation has been recently r e p ~ r t e d . ’ ~ , ~ ~
Compartmental models
The fundamental technique of representing regions as ‘compartments’ has
featured in a wide variety of pharmacokinetic models intended to represent
an equally wide range of kinetic characteristics. This review will discuss only
compartmental models of specific regions or organs of the body, principally
within psychological model^,^^.^' and as specific models of the elimination of
REGIONAL PHARMACOKINETICS
11
drugs by the liver. This does not include the ‘traditional’ compartmental models
of the whole body used in systemic pharmacokinetics.
A compartment may be defined in either a deterministic or stochastic
manner.48 The deterministic definition is that of a pooled region of the body
in which drug is instantaneously mixed and uniformly distributed, while the
stochastic definition is of an homogeneous pool of drug from which all molecules
have an equal probability of entering or leaving.
Flow-limited regional compurtmentul models. In many physiological pharmacokinetic models, a region of the body is represented as one ‘well-stirred’ compartment in which it is assumed that the rate of diffusion of drug across cell
membranes and through the interstitial and intracellular spaces is much more
rapid that the rate of delivery of the drug to the region by the blood flow.
In these flow-limited or ‘well-stirred’ models, the fundamental equation for
describing uptake is derived from mass balance principles, and its most general
form is as follows (adapted from Chen and A ~ ~ d r a d e ~ ~ ) :
.Itol
is the total flux (i.e. mass per unit time) of accumulation of drug in the
compartment; Jinis the flux of inflow of drug into the compartment via the
afferent blood supply; Jinjis the flux of drug administration into compartment
(if any); JOu1
is the flux of outflow of drug via the efferent blood and JCiis
the flux of elimination of the drug from the compartment.
Depending on the application, equation (12) can be expanded using specific
assumptions which can be quite complex. For example, elimination can be
described by Michaelis-Menten kinetics’” while binding within the compartment
may be n ~ n - l i n e a r . ~ ’
and JeIare neglected.’* If the drug concenTo describe drug distribution, Jin,
trations in a compartment of volume V are C,,, and in arterial and venous
blood are C, and C,, respectively, Jlol may be rewritten as (dC,,ldr). V and
Jinand JOu1
as Q . Ca and Q . C,, respectively. To accommodate binding within
the compartment, it is assumed that C, is equal to C,, divided by a partition
coefficient ( R ) .Using these modifications, equation (12) becomes:
The use of a partition coefficient, or the concept of venous equilibrium, has
generally gained wide acceptance for lipophilic drugs. However, studies in
which simultaneous tissue and blood concentrations were measured have shown
both
and bad55.56correlations with the venous equilibrium model.
More sophisticated validation has been possible using isolated perfused liver
12
R. N . UFTON, W. B. RUNCIMAN AND L. E. MATHER
preparations in which perfusate flow and protein binding were controlled, but
it was found that the flow-limited compartment model was not a unique description of the liver for all
Membrane-limited regional compartment models. Although flow-limited compartmental models are widely used for lipophilic drugs, it has been recognized
that the rate of uptake and elution of polar, highly ionized, or charged drugs
may be limited by the rate at which these drug cross either capillary or cell
membranes.32To describe these drugs, complex models of regions have been
proposed with compartments representing the intravascular, interstitial, and
intracellular volumes of the region. Drug transfer between the compartments
is assumed to be limited by the rate at which drug can diffuse across the barriers
separating compartments, and drug may be bound to cellular components,
excreted or metabolized in a linear or non-linear manner within the compartments. GilletteS8published equations describing these types of models, but did
not present the predictions of the equations. Simplified equations for a membrane-limited model are also presented in the review of Gerlowski and J a i r ~ . ~ ~
Of the 64 applications of physiological models reviewed by these authors, only
1 1 included some type of more complex multi-compartmental regional pharmacokinetic model. In general, it appears that the use of these more complex
models has been limited by the lack of evidence that they are necessary which,
in turn, has been limited by the lack of experimental information about the
factors influencing regional drug kinetics, particularly the time-courses of
regional venous, interstitial, and intracellular drug concentrations.
Other regional pharmacokinetic models
Flow-limited, and to a lesser extent membrane-limited, regional compartmental models account for the large majority of regional pharmacokinetic
models. However, there are a number of other models which are not in common
use and have yet to be thoroughly investigated.
Parallel tube59 and distributed modelsm are alternatives to flow-limited
models6' of the liver. In these models, the liver is represented as a number
of parallel tubes representing the hepatic sinusoids, along which hepatic
enzymes are uniformly distributed. This establishes a concentration gradient
from the arterial to the venous ends of the sinusoids. The relative merits of
these models over flow-limited compartment models have generated considerable debate,s7and appear to differ between drugs.
A parallel tube model has been used to describe drug distribution into skeletal
muscles2 using binding sites along the tubes rather than enzymes. It was not
possible to demonstrate that this model was superior to the flow-limited compartment model for describing the distribution of a number of drugs. On current
evidence, it would appear that the representation of a region as a set of parallel
REGIONAL PHARMACOKINETICS
13
tubes lined with either enzymatic or reversible binding sites is not always a
good representation of the in vivo situation, in which these sites are separated
from the blood by both distance and cellular or membrane barriers.
A dispersion model was proposed by Roberts and R ~ w l a n das~ a~ model
of the liver in which the flow-limited compartments and parallel tube models
were subsets, depending on the parameters. The liver was again represented
as a ‘compartment’ in which there was a heterogeneous dispersion of drug.
Although the dispersion model was able to successfully bridge the gap between
these other models of the liver, it has not been applied to other tissues. The
dispersion model is based on principles intended for chemical reactors, and
therefore lacks physiological reality.
Jones and Nicholas62presented a regional pharmacokinetic model of the
lung based on chromatographic principles. The model was stochastic and noncompartmental. In the model, the drug molecules could be in one of two states,
either in the blood stream moving along a length of vasculature at a velocity
determined by the blood flow, or stationary in the tissue of the lung. The
numbers of times a given molecule lodged in the lung tissue was described
by a Poisson distribution, while the length of time a given molecule remained
in the tissue was described by an exponential distribution. The model was
a good representation of the uptake and elution of alphaxolone in the rat
lung.
proposed that many blood concentration-time curves can be
described by gamma distributed residence times, which is analogous to the
chromatographic concept of drug molecules lodging in tissues in their circulation around the body. Models such as these may be useful physiologically
based alternatives to compartmental descriptions of regional pharmacokinetics
which employ exponential functions. If the choice of mathematical equations
to fit to data depends on the physiological basis of the model from which
they were derived, such stochastic methods may have merit.
The most complex regional pharmacokinetic models used to date are those
developed to describe the kinetics of gases,@and those developed by circulation
physiologists to explain the kinetics of tracers.6s These models have yet to
be exploited by pharmacokineticists. An essential feature of both these types
of models is that they are high physiologically realistic and take into account
the limitations of diffusion through a three dimensional section of tissue. The
Krogh cylinder model of oxygen tensions around a capillary was first proposed
in 1919. In this model, the tissue is divided into a number of parallel cylinders
each containing one capillary. The difference in oxygen tension between the
capillary and a point in the cylinder is governed by tissue oxygen consumption,
oxygen solubility, the diffusion coefficient of oxygen and the radius and length
of the cylinder and capillary.66Similar diffusion models based on the model
of Renkid7have been used to describe tracer kinetics, but with the incorporation of parameters representing capillary pe~-meability.~~*~~
Many of these models could be adapted to describe both regional drug
metabolism and distribution. For example, Stec and Atkinsod9 adapted the
14
R. N . UPTON, W. B. RUNCIMAN AND L. E . MATHER
model of R e n k i r ~to~ construct
~
a flow and permeability dependent model of
the regional kinetics of procainamide. It is certain that the use, validation,
and development of these more complex physiological models will be a challenging area of pharmacokinetics.
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