Table Of ContentLectu re Notes in
Biomathematics
Managing Editor: S. Levin
48
Tracer Kinetics
and Physiologic Modeling
Theory to Practice
Proceedings of a Seminar
held at St.Louis, Missouri
June 6, 1983
Edited by R. M. Lambrecht and A. Rescigno
Springer-Verlag Berlin Heidelberg GmbH
Editorial Board
W.Bossert H.J.Bremermann J.D.Cowan W.Hirsch S.Karlin
J.B.Keller M.Kimura S.Levin(Managing Editor)R.C.Lewontin R.May
G.F.Oster A.S.Perelson T.Poggio L.A. Segel
Editors
R.M.Lambrecht
A.Rescigno
Brookhaven National Laboratory/Chemistry Department
Upton, New York11973,USA
AMS SubjectClassification (1980):92-XX
Thisworkissubjecttocopyright. Allrightsarereserved,whetherthewholeorpartofthematerial
isconcerned, specificallythoseoftranslation,reprinting,re-useofillustrations,broadcasting,
reproductionbyphotocopyingmachineorsimilar means,andstorageindatabanks.Under
§54oftheGermanCopyrightLawwherecopies aremadeforother thanprivateuse,afeeis
payableto"VerwertungsgesellschaftWort",Munich.
ISBN 978-3-540-12322-2 ISBN 978-3-642-50036-7 (eBook)
DOI 10.1007/978-3-642-50036-7
©bySpringer-Verlag BerlinHeidelberg 1983
OriginallypublishedbySpringer-VerlagBerlinHeidelberg in1983.
2146/3140-543210
PREFACE
These lectures were prepared by the authors for Seminars to
be held on June 6, 1983, in St.Louis, Missouri, under the spon
sorship of the Radiopharmaceutical Science Council of the So
ciety of Nuclear Medicine. All manuscript? were refereed.
Tracer kinetics and the modeling of physiological and bio
chemical processes in vivo are the focus of a contemporary di
rection in biochemical research. Recent advances in instrumen
tation (especially positron emission tomography and digital
autoradiography) and parallel developments in the production of
short-lived radionuclides and rapid synthetic chemistry to pre
pare tracers that probe metabolism, flow, receptor-ligand
kinetics, etc., are responsible for new scientific frontiers.
These developments, coupled with biomathematics and computer
science, make it possible to quantitatively evaluate tracer
kinetic models in animals and man. (The choice of animal models
in radiotracer design and tracer kinetics is the subject of a
book edited by R.M.Lambrecht and W.C.Eckleman in press at Sprin
ger Verlag.)
Tracer kinetics and physiological modeling is truly multi
disciplinary, as evidenced by the intellectual diversity and
international representation observable in the list of contri
butors. The lectures outline and attempt to show the transition
from the theoretical description to the practical application
of modeling for understanding normal and pathological processes.
It is our hope that young researchers, particularly in the
physical, radiopharmaceutical, and medical sciences, will be
inspired to apply tracer kinetics and physiological modeling in
their work;
Quogue, NY Richard M.Lambrecht
December 31, 1982 Aldo Rescigno
TABLE OF CONTENTS
THE STATISTICAL ANALYSIS OF PHARMACOKINETIC DATA
James H. Matis, Thomas E. Wehrly, and Kenneth B.
Gerald
MATHEMATICAL METHODS IN THE FORMULATION OF PHARMACOKINETIC
MODELS 59
Aldo Rescigno, Richard M. Lambrecht, and Charles C.
Duncan
NEW APPROACHES TO UPTAKE BY HETEROGENEOUS PERFUSED ORGANS:
FROM LINEAR TO SATURATION KINETICS 120
Ludvik Bass, Anthony J. Bracken, and Conrad J. Burden
BASIC PRINCIPLES UNDERLYING RADIOISOTOPIC METHODS FOR ASSAY
OF BIOCHEMICAL PROCESSES IN VIVO 202
Louis Sokoloff and Carolyn B. Smith
TRACER STUDIES OF PERIPHERAL CIRCULATION 235
Niels A. Lassen and Ole Henriksen
TRACER KINETIC MODELING IN POSITRON COMPUTED TOMOGRAPHY 298
Sung-Cheng Huang, Richard E. Carson, and Michael E.
Phelps
PHARMACOKINETIC MODELS AND POSITRON EMISSION TOMOGRAPHY:
STUDIES OF PHYSIOLOGIC AND PATHOPHYSIOLOGIC CONDITIONS 345
Martin Reivich and Abass Alavi
KINETIC ANALYSIS OF THE UPTAKE OF GLUCOSE AND SOME OF ITS
ANALOGS IN THE BRAIN USING THE SINGLE CAPILLARY MODEL: 348
COMMENTS ON SOME POINTS OF CONTROVERSY
Niels A. Lassen and Albert Gjedde
THE USE OF llC-METHYL-D-GLUCOSE FOR ASSESSMENT OF GLUCOSE
TRANSPORT IN THE HUMAN BRAIN: THEORY AND APPLICATION 403
Karl Vyska, Miroslav Profant, Franz Schuier,
C. Freundlieb, Anton Hock, Hans-U Thal, Veit Becker,
Ludwig E. Feinendegen
THE INDICATOR DILUTION METHOD: ASSUMPTIONS AND
APPLICATIONS TO BRAIN UPTAKE 429
Olaf B. Paulson and Marianne M. Hertz
MEASUREMENT OF LIGAND-RECEPTOR BINDING: THEORY AND
PRACTICE 445
David E. Schafer
ACKNOWLEDGEMENTS 509
LIST OF CONTRIBUTORS
Alavi, Abass Departments of Neurology and Nuclear
Medicine, University of Pennsylvania
Hospital,· Philadelphia, PA 19104
Bass, Ludvik Department of Mathematics, University
of Queensland, St. Lucia, Queensland,
Australia 4067
Becker, Veit Institute of Medicine, Nuclear
Research Center, JUlich, and
Department of Nuclear Medicine,
University of Dusseldorf,
5170 JUlich, F.R. Germany
Bracken, Anthony J. Department of Mathematics, University
of Queensland, St. Lucia, Queensland,
Australia 4067
Burden, Conrad J. Department of Nuclear Physics,
Weizmann Institute of Science,
Rehovot, Israel
Carson, Richard E. Division of Biophysics, Department of
Radiological Sciences and Laboratory
of Nuclear Medicine, UCLA School of
Medicine, Los Angeles, CA 90024
Duncan, Charles C. Section of Neurosurgery, Yale
University School of Medicine, New
Haven, CT 06510
Feinendegen, Ludwig Emil Institute of Medicine, Nuclear
Research Center, Julich, and
Department of Nuclear Medicine,
University of Dusseldorf,
5170 JUlich, F.R. Germany
Freundlietl, Christian Institute of Medicine, Nuclear
Research Center, JUlich, and
Department of Nuclear Medicine,
University of Dusseldorf,
5170 JUlich, F.R. Germany
Gerald, Kenneth B. Department of Biometry, University of
Kansas Medical Center, Kansas City,
KS 66103
Gjedde, Albert Department of Clinical Physiology,
Bispebjerg Hospital. DK-2100
Copenhagen, Denmark
Henriksen, Ole Department of Clinical Physiology,
Bispebjerg Hospital. DK-2100
Copenhagen. Denmark
Hertz, Marianne M. Department of Psychiatry.
Rigshospitalet, State University
Hospital, DK-2100 Copenhagen, Denmark
Hock, Anton Institute of Medicine, Nuclear
Research Center, JUlich, and
Department of Nuclear Medicine,
University of Dusseldorf.
5170 JUlich, F.R. Germany
Huang, Sung-Cheng Division of Biophysics. Department of
Radiological Sciences and Laboratory
of Nuclear Medicine, UCLA School of
Medicine. Los Angeles, CA 90024
Lambrecht, Richard M. Chemistry Department. Brookhaven
National Laboratory. Upton, NY 11973
Lassen, Niels A. Department of Clinical Physiology,
Bispebjerg Hospital. DK-2100
Copenhagen. Denmark
Matis, James H Institute of Statistics, Texas A&M
University. College Station, TX
77840
Paulson, Olaf B. Department of Neurology.
Rigshospitalet. State University
Hospital. DK-2100 Copenhagen, Denmark
Phelps, Michael E. Division of Biophysics, Department of
Radiological Sciences 'and Laboratory
of Nuclear Medicine. UCLA School of
Medicine. Los Angeles. CA 90024
Profant, Miroslav Department of Applied Mathematics.
University of Duisburg,
4100 Duisburg. F.R. Germany
Reivich, Martin Department of Neurology. University
of Pennsylvania Hospital,
Philadelphia. PA 19104
VIII
Rescigno, Aldo Chemistry Department, Brookhaven
National Laboratory, Upton, NY 11973
and Yale University, New Haven" CT
06510
Schafer, David E. Veterans Administration Medical
Center, West Haven, CT 06516
Schuier, Franz Department of Neurology, University
of Dusseldorf, 4000 Dusseldorf,
F.R. Germany
Smith, Carolyn B. Laboratory of Cerebral Metabolism,
National Institute of Mental Health,
Bethesda, MD 20205
Sokoloff, Louis Laboratory of Cerebral Metabolism,
National Institute of Mental Health,
Bethesda, MD 20205
Thal, Hans-Uwe Department of Neurosurgery,
University of Dusseldorf,
4000 Dusseldorf, F.R. Germany
Vyska, Karl Institute of Medicine, Nuclear
Research Center, JUlich, and
Department of Nuclear Medicine,
University of Dusseldorf,
5170 JUlich, F.R. Germany
Wagner, Henry N. Jr. The Johns Hopkins Medical
Institutions, Divisions of Nuclear
Medicine and Radiation Health
Sciences, Baltimore, MD 21205
Wehrly, Thomas E. Institute of Statistics, Texas A&M
University, College Station, TX
77840
THE STATISTICAL ANALYSIS OF PHARMACOKINETIC DATA
James H. Matis*, Thomas E. Wehrly*, Kenneth B. Gerald**
*Texas A&M University, College Station, TX; **University of
Kansas Medical Center, Kansas City, Kansas
1. INTRODUCTION
Virtually all of the practical applications of pharmacokinetic
data analysis involve two parts, one is the mathematical modelling
of the underlying pharmacokinetic system and the other is the
statistical analysis of pharmacokinetic data. The mathematical
modelling has receiv~d great attention in the literature and has
produced many practically useful and mathematically elegant models.
The mathematical modelling of compartmental systems is described
in (1-6). However, the statistical analysis of pharmacokinetic
data has received relatively little attention in the literature.
The limited attention that it has received usually addresses one
of the following questions.
1) What kind of measurement error should one attach to
the given mathematical model, or in other words,how
should one weight the data?
2) What are the statistical techniques to fit the given
mathematical, typically deterministic models to the data?
Implicit in both of these questions is a separation of the modelling
process from the subsequent statistical analysis.
This chapter presents an attempt to integrate the modelling
and the subsequent statistical analysis. Section 2 outlines the
classical deterministic model with measurement error, and it
reviews in some detail its associated statistical analysis.
The section adds some recent developments of practical interest
and may be regarded as a supplement to ( 7). Section 3 presents
a stochastic foundation for the model. The inherent process
error is derived for the model and incorporated into the statis
tical analysis. Section 4 and 5 first derive models with
alternative stochastic causal mechanisms and then discuss the
statistical analysis associated with such models. Section 6
derives estimates of various residence time moments and formulates
a statistical analysis based on such transformed variables. Each
section is structured first to derive the model and its inherent
error distribution, and then to indicate an appropriate statisti
cal analysis for the model with its assumed error structure.
2
The statistical analysis of stochastic models is a natural
extension of previous research on the analysis of deterministic
models. It is a very promising and emerging area for both the
researcher and the practitioner. Of course, the statistical analy
sis of any real-world data set may be considered as much an art as
a science, and in this light the chapter may raise more questions
than it answers. However, we are not aware of any previous compre
hensive review of the statistical analysis of these stochastic
compartmental models. Hopefully this review will serve as a
significant first step in this area and will encourage many further
developments.
2. STATISTICAL ANALYSIS OF DATA FROM DETERMINISTIC MODELS
2.1 Deterministic Formulation of Model
Rescigno and Segre (4 ) out1ine the history of the develop
ment of the deterministic compartmental model, and some recent
developments are noted in Rescigno (8) (Chapter 1 of this book).
These linear system models have been solved historically using
the theory of Laplace transforms. However, for subsequent con
venience, the formulation below will be presented in a matrix
framework and then solved using matrix analysis. This formulation,
fromMatis et a1. (9), is unique in that it models the system
separately for each compartment of origin. This approach will be
useful later in tying together various model assumptions.
Consider the following definitions:
1) Let Xij(t), i ,j=l ... , n; denote the amount of sub
stance in compartment j at time t that originated in
compartment i either at the initial time, t=O, or by
subsequent entry into i.
2) Let k.o, i=l, ... ,n; j=O, 1, ... ,n; with itj;
lJ f . C
denote the nonnegative transfer rate rom1 to J0. om-
partment 0 denotes the system exterior. The units are
1.
time-
n
I
3) Let k k ·, i=l, ... , n; be the total output
ii iJ
j=O
Hi
rate from i .
4) Let fOi(t), i=l, . .. , n; be the nonnegative input
rate at time t from the system exterior to i. The units
are mass/time. • •
5) Let ~(t) = [Xi .(t)] and ~(t) = [Xij(t)] be the nxn matrices
J
of amounts ana their derivatives at time t , respectively.
6) Let K= [k.o] and F(t) = diag [fOi(t)] be the nxn matrix
- lJ -
of flow rates and the nxn diagonal matrix of input rates
at time t, respectively.
3
7) Let ~ = diag [~i] and T = [Il' . . . , In] be, respec
tively, the diagonal matrix of eigenvalues of Kand a
corresponding matrix of right eigenvectors of K. By
definition, one has ~I = I~
The usual (linear) compartment model may be represented by the
following nxn system of linear differential equations:
g(t) = ~(t)~ + E(t) (1)
Let us now assume the following regularity conditions for the
mode1 in Eq. 1.
(i) Each compartment is open, i.e., for each i > 1 there
exists some j > 0 such that k,.. > O. -
- J
(ii) The system is open to the exterior, i.e., there exists
some i ::.1 such that kiO> 0 .
(iit) The system is at least weakly connected, i.e., the
system cannot be partitioned into two sets 51 and 52
such that k.. = 0 = k.. for all compartments i in 51
and j in 52~J J'
(iv) The eigenvalues, Ai are distinct.
Most biomedical system models satisfy conditions (i) to (iii)
either directly or through slight model redefinitions. For
example, if a system has a closed compartment, also called a sink,
the closed compartment may be redefined as an exterior state and
thus satisfy (i). Asystem which is closed to the exterior will
follow the subsequent theory with slight modifi cations to
accomodate its single eigenvalue of O. Asystem which is not at
least weakly connected may be partitioned into two mutually
exclusive subsystems, and each subsystemmay be analyzed separately.
Models with equal eigenvalues in violation of (iv) are not of
present interest for statistical analysis since, unless constrained
otherwise, the estimates obtained by subsequent data analysis will
differ with probability 1. Hence regularity conditions (i) to
(iv) are reasonable to impose on system models.
The following theorem has been proven for such linear systems:
Theorem 1: Let e~t = diag(eAit). The solution to Eq. 1 with
regularity conditions (i) - (iv) is
a) for fOi(t) = 0, i=l, ... , n;
~(t) = ~(O) Te~tT-1 (2)
b) for Xii(O) 0, i=l, ... , n;
~(t) ~(s) Ie~(t-s) r '
= Jt ds (3)
