Tracer Kinetics in Biomedical Research: From Data to Model

469 Pages·2002·8.248 MB·English

by  Cobelli C., Foster D., Toffolo G.

Tracer Kinetics in
Biomedical Research
This page intentionally left blank.
Tracer Kinetics in
Biomedical Research
From Data to Model
Claudio Cobelli
University of Padova
Padova, Italy
David Foster
University of Washington
Seattle, Washington
and
Gianna Toffolo
University of Padova
Padova, Italy
KLUWER ACADEMIC PUBLISHERS
NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW
eBookISBN: 0-306-46833-6
Print ISBN: 0-306-46427-6
©2002 Kluwer Academic Publishers
New York, Boston, Dordrecht, London, Moscow
All rights reserved
No part of this eBook may be reproduced or transmitted in any form or by any means, electronic,
mechanical, recording, or otherwise, without written consent from the Publisher
Created in the United States of America
Visit Kluwer Online at: http://www.kluweronline.com
and Kluwer's eBookstore at: http://www.ebooks.kluweronline.com
Preface
The use of mathematical modeling techniques in biomedical research
is playing an increasingly important role as one seeks to understand the
physiopathology of disease processes. This includes not only understand-
ing mechanisms of physiological processes, but diagnosis and treatment.
In addition, its introduction in the study of genomics and proteomics
is key in understanding the functional characteristics of gene expression
and protein assembly and secretion. Finally, with the increasing com-
plexity and associated cost of drug development, modeling techniques
are being used to streamline the process.
We have worked in close collaboration with colleagues in biomedical
and pharmaceutical research for a number of years applying and refining
mathematical modeling techniques to a variety of problems. In addition,
we have worked in collaboration with colleagues in applied mathematics
and statistics to develop new algorithms to solve new sets of problems
as they emerge in our research efforts. Finally, we have worked with
colleagues in computer science to develop new software tools that bring
the power of mathematical modeling to a broad research community.
This books brings together much of what we have learned over the years,
and presents the material in a format that should be accessible both to
the novice reader and those desiring more detailed information about
specific techniques.
We are indebted to many of our colleagues who were extremely patient
and helpful during the preparation of the book for publication. We
are encouraged by the support we have received from our respective
institutions and also review panels for several of the research grants we
have obtained during the work on the book.
There are many research programs that have led directly to material
presented in the text. Special mention must be given to the Biomedical
Technology Program in the National Center for Research Resources at
v
vi TRACER KINETICS IN BIOMEDICAL RESEARCH
the National Institutes of Health (USA) whose resource facility grant
Resource Facility for Kinetic Analysis (RFKA) supported all authors
during the development of the SAAM II software system. There is a
tight link between the material developed in this text and SAAM II;
SAAM II was used to develop all examples in the text.
The preparation of the book would not have been possible without
regular travel between Seattle and Padova. Funding for the travel was
provided by RFKA and the Ministero della Università e Ricerca Scien-
tifica e Tecnologica of Italy. We are most grateful for this support.
Finally, we would like to thank Agnes Sieger and Mike Macaulay for
the final preparation of the text.
CLAUDIO CODELLI, PADOVA, ITALY
DAVID FOSTER, SEATTLE, WASHINGTON, USA
GlANNA TOFFOLO, PADOVA, ITALY
Contents
1. INTRODUCTION 1
1.1 Why Modeling? 1
1.2 How Modeling? 3
1.3 Aim of the Book 5
1.4 Who Should Read the Book? 5
1.5 Organization of the Book 6
2. FUNDAMENTALS OF TRACER KINETICS 11
2.1 Introduction 11
2.2 The Tracer-Tracee System 12
2.2.1 Concepts and definitions 12
2.2.2 The tracee system 15
2.2.3 The tracer system 17
2.2.4 The tracer-tracee system 19
2.2.5 System parameters from tracer and tracee
measurements 19
2.3 The Tracer-Tracee System with Isotopic Tracers 21
2.3.1 Concepts and definitions 21
2.3.2 Relationships among isotopic variables 24
2.4 The Radioactive Tracer Variables 25
2.4.1 Measurements 25
2.4.2 Kinetic variables 27
2.5 The Stable Isotope Tracer Variables 28
2.5.1 Measurements 28
2.5.2 Kinetic variables 31
2.5.3 The multiple species case 33
2.5.4 A test of the endogenous constant
assumption 35
2.6 Multiple Tracer Experiments 37
3. THE NONCOMPARTMENTAL MODEL OF MULTIPOOL
SYSTEMS: ACCESSIBLE POOL AND SYSTEM PARMETERS 39
3.1 From Single to Multipool Systems 39
vii
viii TRACER KINETICS IN BIOMEDICAL RESEARCH
3.2 Kinetic Parameters of the Accessible Pool 43
3.2.1 Definitions 43
3.2.2 Formulas 44
3.3 Kinetic Parameters of the System 51
3.3.1 Definitions 51
3.3.2 Formulas 52
3.3.3 Limitations of noncompartmental models 56
3.3.4 Parameters from total body tracer measurements 58
3.4 The Two Accessible Pool Noncompartmental Model:
Accesssible Pool and System Kinetic Parameters 59
3.4.1 Introduction 59
3.4.2 The two input - four output experiment for
radioactive and stable isotope tracers 62
3.4.3 The accessible pool kinetic parameters: Definitions
and formulas 64
3.4.4 System parameters: Definitions and formulas 69
3.4.5 Relationship between one and two accessible pool
noncompartmental models 71
3.4.6 Limitations of two accessible pool noncompart-
mental models 72
4. THE COMPARTMENTAL MODEL 75
4.1 Introduction 75
4.2 Concepts and Definitions 77
4.3 The Compartmental Model of a Tracer-Tracee System 80
4.3.1 Introduction 80
4.3.2 The one compartment model 82
4.3.3 The two compartment model 84
4.3.4 The n-compartment model 86
4.4 Structural Properties 91
4.4.1 Introduction 91
4.4.2 The compartmental matrix 91
4.4.3 The mean residence time matrix 94
4.4.4 Sums of exponentials and the compartmental
model 97
4.4.5 Non-negativity and stability properties of
compartmental model equations 99
4.5 Kinetic Parameters 103
4.6 Catenary and Mammillary Models 105
5. IDENTIFIABILITY OF THE TRACER MODEL 109
5.1 Introduction 109
5.2 Some Examples 110
5.3 Definitions 118
5.4 The Two Compartment Model 120
5.4.1 Introduction 120
5.4.2 Input into a single compartment 121
5.4.3 Input into both compartments 129
Contents ix
5.5 The Laplace Transform Method: The Two Compartment
Model Revisited 132
5.5.1 Introduction 132
5.5.2 Example of the Laplace transform method 133
5.6 The Difficulty of the Identifiability Problem 135
5.7 The Three Compartment Model 139
5.8 Catenary and Mammillary Models 143
5.9 A Priori Identifiability of General Structure Compart-
mental Models: A Computer Algebra Approach 148
5.9.1 Introduction 148
5.9.2 Rationale 151
5.9.3 The transfer function topological approach 151
5.9.4 The identifiability algorithm 153
6. USING THE TRACER MODEL TO ESTIMATE KINETIC
PARAMETERS 165
6.1 Introduction 165
6.2 Estimation from A Priori Uniquely Identifiable Models 168
6.3 Estimation from Interval Identifiable Models 179
COMPARTMENTAL VERSUS NONCOMPARTMENTAL
KINETIC PARAMETERS 191
7.1 Introduction 191
7.2 The Mean Residence Time Matrix Revisited 192
7.3 Equivalence of the Accessible Pool Parameters 194
7.4 Nonequivalence of the System Parameters 200
7.5 Parameters of the Nonaccessible Pools 207
7.6 The Two Accessible Pool Model 208
7.6.1 Accessible pool parameters 208
7.6.2 System parameters 209
8. PARAMETER ESTIMATION: SOME FUNDAMENTALS
OF REGRESSION ANALYSIS 215
8.1 Introduction 215
8.1.1 The nature of the regression problem 216
8.1.2 Linear and nonlinear parameters 220
8.2 Basic Concents of Regression Analysis 221
8.2.1 The residual 221
8.2.2 Residual sum of squares 222
8.2.3 Weights and weighted residual sum of squares 224
8.3 The Assignment of Weights to Data 225
8.3.1 Introduction 225
8.3.2 Description of the error in the data 226
8.3.3 Weights and error variances 227
8.3.4 A model of the error variance 228
8.3.5 Estimating the parameters of the error model from
standard samples 231