Macheras-FM.qxd 11/17/05 9:36 AM Page i
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`Interdisciplinary Applied Mathematics
`
`Volume 30
`
`Editors
`S.S. Antman J.E. Marsden
`L. Sirovich S. Wiggins
`
`Geophysics and Planetary Sciences
`
`Mathematical Biology
`L. Glass, J.D. Murray
`
`Mechanics and Materials
`R.V. Kohn
`
`Systems and Control
`S.S. Sastry, P.S. Krishnaprasad
`
`Problems in engineering, computational science, and the physical and biological
`sciences are using increasingly sophisticated mathematical techniques. Thus, the
`bridge between the mathematical sciences and other disciplines is heavily traveled.
`The correspondingly increased dialog between the disciplines has led to the estab-
`lishment of the series: Interdisciplinary Applied Mathematics.
`
`The purpose of this series is to meet the current and future needs for the interaction
`between various science and technology areas on the one hand and mathematics on
`the other. This is done, firstly, by encouraging the ways that that mathematics may be
`applied in traditional areas, as well as point towards new and innovative areas of
`applications; and, secondly, by encouraging other scientific disciplines to engage in a
`dialog with mathematicians outlining their problems to both access new methods
`and suggest innovative developments within mathematics itself.
`
`The series will consist of monographs and high-level texts from researchers working
`on the interplay between mathematics and other fields of science and technology.
`
`Page 1
`
`SHIRE EX. 2035
`KVK v. SHIRE
`IPR2018-00290
`
`

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`Interdisciplinary Applied Mathematics
`Volumes published are listed at the end of the book
`
`Page 2
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`
`Panos Macheras
`
`Athanassios Iliadis
`
`Modeling in
`Biopharmaceutics,
`Pharmacokinetics, and
`Pharmacodynamics
`Homogeneous and Heterogeneous
`Approaches
`
`With 131 Illustrations
`
`Page 3
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`

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`
`Panos Macheras
`School of Pharmacy
`Zographou 15771
`Greece
`Macheras@pharm.uoa.gr
`
`Athanassios Iliadis
`Faculty of Pharmacy
`Marseilles 13385 CX 0713284
`France
`Iliadis@pharmacie.univ-mrs.fr
`
`J.E. Marsden
`Series Editors
`Control and Dynamical Systems
`S.S. Antman
`Mail Code 107-81
`Department of Mathematics and
`Institute for Physical Science and Technology California Institute of Technology
`University of Maryland
`Pasadena, CA 91125
`USA
`College Park, MD 20742
`marsden@cds.caltech.edu
`USA
`ssa@math.umd.edu
`
`L. Sirovich
`Laboratory of Applied Mathematics
`Department of Biomathematics
`Mt. Sinai School of Medicine
`Box 1012
`NYC 10029
`USA
`
`S. Wiggins
`School of Mathematics
`University of Bristol
`Bristol BS8 1TW
`UK
`s.wiggins@bris.ac.uk
`
`Cover illustration: Left panel: Stochastic description of the kinetics of a population of particles,
`Fig 9.15. Middle panel: Dissolution in topologically restricted media, Fig. 6.8B (reprinted with
`permission from Springer). Right panel: A pseudophase space for a chaotic model of cortisol
`kinetics, Fig.11.11.
`
`Mathematics Subject Classification (2000): 92C 45 (main n°), 62P10, 74H65, 60K20.
`
`Library of Congress Control Number: 2005934524
`
`ISBN-10: 0-387-28178-9
`ISBN-13: 978-0387-28178-0
`© 2006 Springer Science+Business Media, Inc.
`All rights reserved. This work may not be translated or copied in whole or in part without the
`written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street,
`New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly
`analysis. Use in connection with any form of information storage and retrieval, electronic adap-
`tation, computer software, or by similar or dissimilar methodology now known or hereafter
`developed is forbidden.
`The use in this publication of trade names, trademarks, service marks, and similar terms, even
`if they are not identified as such, is not to be taken as an expression of opinion as to whether
`or not they are subject to proprietary rights.
`
`While the advice and information in this book are believed to be true and accurate at the date
`of going to press, neither the authors nor the editors nor the ublisher can accept any legal
`responsibility for any errors or omissions that may be made. The publisher makes no warranty,
`express or implied, with respect to the material contained herein.
`
`Printed in the United States of America.
`
`(MVY)
`
`9 8 7 6 5 4 3 2 1
`
`springeronline.com
`
`Page 4
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`♦ To our ancestors who inspired us
`♦ To those teachers who guided us
`♦ To our families
`
`Page 5
`
`

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`Interdisciplinary Applied Mathematics
`
`4.
`
`1. Gutzwiller: Chaos in Classical and Quantum Mechanics
`2. Wiggins: Chaotic Transport in Dynamical Systems
`3.
`Joseph/Renardy: Fundamentals of Two-Fluid Dynamics: Part I:
`Mathematical Theory and Applications
`Joseph/Renardy: Fundamentals of Two-Fluid Dynamics: Part II:
`Lubricated Transport, Drops and Miscible Liquids
`5. Seydel: Practical Bifurcation and Stability Analysis: From Equilibrium to
`Chaos
`6. Hornung: Homogenization and Porous Media
`7. Simo/Hughes: Computational Inelasticity
`8. Keener/Sneyd: Mathematical Physiology
`9. Han/Reddy: Plasticity: Mathematical Theory and Numerical Analysis
`10. Sastry: Nonlinear Systems: Analysis, Stability, and Control
`11. McCarthy: Geometric Design of Linkages
`12. Winfree: The Geometry of Biological Time (Second Edition)
`13. Bleistein/Cohen/Stockwell: Mathematics of Multidimensional Seismic
`Imaging, Migration, and Inversion
`14. Okubo/Levin: Diffusion and Ecological Problems: Modern Perspectives
`(Second Edition)
`15. Logan: Transport Modeling in Hydrogeochemical Systems
`16. Torquato: Random Heterogeneous Materials: Microstructure and
`Macroscopic Properties
`17. Murray: Mathematical Biology I: An Introduction (Third Edition)
`18. Murray: Mathematical Biology II: Spatial Models and Biomedical
`Applications (Third Edition)
`19. Kimmel/Axelrod: Branching Processes in Biology
`20. Fall/Marland/Wagner/Tyson (Editors): Computational Cell Biology
`21. Schlick: Molecular Modeling and Simulation: An Interdisciplinary Guide
`22. Sahimi: Heterogeneous Materials: Linear Transport and Optical Properties
`(Vol. I)
`23. Sahimi: Heterogeneous Materials: Nonlinear and Breakdown Properties
`and Atomistic Modeling (Vol. II)
`24. Bloch: Nonholonomic Mechanics and Control
`25. Beuter/Glass/Mackey/Titcombe: Nonlinear Dynamics in Physiology and
`Medicine
`26. Ma/Soatto/Kosecka/Sastry: An Invitation to 3-D Vision
`27. Ewens: Mathematical Population Genetics (2nd Edition)
`28. Wyatt: Quantum Dynamics with Trajectories
`29. Karniadakis: Microflows and Nanoflows
`30. Macheras/Iliadis: Modeling in Biopharmaceutics, Pharmacokinetics, and
`Pharmacodynamics: Homogeneous and Heterogeneous Approaches
`
`Page 6
`
`

`

`Preface
`
`H µεγ ´αλη τ´εχνη βρ´ισκετ αι oπoυδ´ηπoτ ε o ´ανθρωπoς κατ oρθ ´ωνει
`ν(cid:1)αναγνωρ´ιζει τ oν εαυτ ´oν τ oυ και να τ oν εκϕρ´αζει µε πληρ´oτ ητ α
`µες στ o ελ´αχιστ o.
`
`Great art is found wherever man achieves an understanding of self
`and is able to express himself fully in the simplest manner.
`Odysseas Elytis (1911-1996)
`1979 Nobel Laureate in Literature
`The magic of Papadiamantis
`
`Biopharmaceutics, pharmacokinetics, and pharmacodynamics are the most
`important parts of pharmaceutical sciences because they bridge the gap between
`the basic sciences and the clinical application of drugs. The modeling approaches
`in all three disciplines attempt to:
`• describe the functional relationships among the variables of the system
`under study and
`• provide adequate information for the underlying mechanisms.
`
`Due to the complexity of the biopharmaceutic, pharmacokinetic, and phar-
`macodynamic phenomena, novel physically physiologically based modeling ap-
`proaches are sought. In this context, it has been more than ten years since we
`started contemplating the proper answer to the following complexity-relevant
`questions: Is a solid drug particle an ideal sphere? Is drug diffusion in a well-
`stirred dissolution medium similar to its diffusion in the gastrointestinal fluids?
`Why should peripheral compartments, each with homogeneous concentrations,
`be considered in a pharmacokinetic model? Can the complexity of arterial and
`venular trees be described quantitatively? Why is the pulsatility of hormone
`plasma levels ignored in pharmacokinetic-dynamic models? Over time we real-
`ized that questions of this kind can be properly answered only with an intuition
`about the underlying heterogeneity of the phenomena and the dynamics of the
`processes. Accordingly, we borrowed geometric, diffusional, and dynamic con-
`cepts and tools from physics and mathematics and applied them to the analysis
`of complex biopharmaceutic, pharmacokinetic, and pharmacodynamic phenom-
`ena. Thus, this book grew out of our conversations with fellow colleagues,
`
`vii
`
`Page 7
`
`

`

`viii
`
`Preface
`
`correspondence, and joint publications. It is intended to introduce the concepts
`of fractals, anomalous diffusion, and the associated nonclassical kinetics, and
`stochastic modeling, within nonlinear dynamics and illuminate with their use
`the intrinsic complexity of drug processes in homogeneous and heterogeneous
`media. In parallel fashion, we also cover in this book all classical models that
`have direct relevance and application to the biopharmaceutics, pharmacokinet-
`ics, and pharmacodynamics.
`The book is divided into four sections, with Part I, Chapters 1—3, presenting
`the basic new concepts: fractals, nonclassical diffusion-kinetics, and nonlinear
`dynamics; Part II, Chapters 4—6, presenting the classical and nonclassical mod-
`els used in drug dissolution, release, and absorption; Part III, Chapters 7—9,
`presenting empirical, compartmental, and stochastic pharmacokinetic models;
`and Part IV, Chapters 10 and 11, presenting classical and nonclassical phar-
`macodynamic models. The level of mathematics required for understanding
`each chapter varies. Chapters 1 and 2 require undergraduate-level algebra and
`calculus. Chapters 3—8, 10, and 11 require knowledge of upper undergraduate
`to graduate-level linear analysis, calculus, differential equations, and statistics.
`Chapter 9 requires knowledge of probability theory.
`We would like now to provide some explanations in regard to the use of
`some terms written in italics below, which are used extensively in this book
`starting with homogeneous vs. heterogeneous processes. The former term refers
`to kinetic processes taking place in well-stirred, Euclidean media where the
`classical laws of diffusion and kinetics apply. The term heterogeneous is used
`for processes taking place in disordered media or under topological constraints
`where classical diffusion-kinetic laws are not applicable. The word nonlinear
`is associated with either the kinetic or the dynamic aspects of the phenomena.
`When the kinetic features of the processes are nonlinear, we basically refer to
`Michaelis—Menten-type kinetics. When the dynamic features of the phenomena
`are studied, we refer to nonlinear dynamics as delineated in Chapter 3.
`A process is a real entity evolving, in relation to time, in a given environment
`under the influence of internal mechanisms and external stimuli. A model is an
`image or abstraction of reality: a mental, physical, or mathematical represen-
`tation or description of an actual process, suitable for a certain purpose. The
`model need not be a true and accurate description of the process, nor need the
`user have to believe so, in order to serve its purpose. Herein, only mathematical
`models are used. Either processes or models can be conceived as boxes receiv-
`ing inputs and producing outputs. The boxes may be characterized as gray or
`black, when the internal mechanisms and parameters are associated or not with
`a physical interpretation, respectively. The system is a complex entity formed
`of many, often diverse, interrelated elements serving a common goal. All these
`elements are considered as dynamic processes and models. Here, determinis-
`tic, random, or chaotic real processes and the mathematical models describing
`them will be referenced as systems. Whenever the word “system” has a specific
`meaning like process or model, it will be addressed as such.
`For certain processes, it is appropriate to describe globally their properties
`using numerical techniques that extract the basic information from measured
`
`Page 8
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`

`

`Preface
`
`ix
`
`data. In the domain of linear processes, such techniques are correlation analysis,
`spectral analysis, etc., and in the domain of nonlinear processes, the correlation
`dimension, the Lyapunov exponent, etc. These techniques are usually called
`nonparametric models or, simply, indices. For more advanced applications, it
`may be necessary to use models that describe the functional relationships among
`the system variables in terms of mathematical expressions like difference or dif-
`ferential equations. These models assume a prespecified parametrized structure.
`Such models are called parametric models.
`Usually, a mathematical model simulates a process behavior, in what can
`be termed a forward problem. The inverse problem is, given the experimental
`measurements of behavior, what is the structure? A difficult problem, but an
`important one for the sciences. The inverse problem may be partitioned into the
`following stages: hypothesis formulation, i.e., model specification, definition of
`the experiments, identifiability, parameter estimation, experiment, and analysis
`and model checking. Typically, from measured data, nonparametric indices are
`evaluated in order to reveal the basic features and mechanisms of the underlying
`processes. Then, based on this information, several structures are assayed for
`candidate parametric models. Nevertheless, in this book we look only into
`various aspects of the forward problem: given the structure and the parameter
`values, how does the system behave?
`Here, the use of the term “model” follows Kac’s remark, “models are cari-
`catures of reality, but if they are good they portray some of the features of the
`real world” [1]. As caricatures, models may acquire different forms to describe
`the same process. Also, Fourier remarked, “nature is indifferent toward the dif-
`ficulties it causes a mathematician,” in other words the mathematics should be
`dictated by the biology and not vice versa. For choosing among such compet-
`ing models, the “parsimony rule,” Occam’s “razor rule,” or Mach’s “economy
`of thought” may be the determining criteria. Moreover, modeling should be
`dependent on the purposes of its use. So, for the same process, one may de-
`velop models for process identification, simulation, control, etc. In this vein,
`the tourist map of Athens or the system controlling the urban traffic in Mar-
`seilles are both tools associated with the real life in these cities. The first is an
`identification model, the second, a control model.
`Over the years we have benefited enormously from discussions and collab-
`orations with students and colleagues.
`In particular we thank P. Argyrakis,
`D. Barbolosi, A. Dokoumetzidis, A. Kalampokis, E. Karalis, K. Kosmidis, C.
`Meille, E. Rinaki, and G. Valsami. We wish to thank J. Lukas whose suggestions
`and criticisms greatly improved the manuscript.
`
`A. Iliadis
`Marseilles, France
`August 2005
`
`P. Macheras
`Piraeus, Greece
`August 2005
`
`Page 9
`
`

`

`Contents
`
`Preface
`
`List of Figures
`
`I BASIC CONCEPTS
`
`1 The Geometry of Nature
`1.1 Geometric and Statistical Self-Similarity . . . . . . . . . . . . . .
`1.2 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
`1.3 Fractal Dimension . . . . . . . . . . . . . . . . . . . . . . . . . .
`1.4 Estimation of Fractal Dimension . . . . . . . . . . . . . . . . . .
`1.4.1
`Self-Similarity Considerations . . . . . . . . . . . . . . . .
`1.4.2 Power-Law Scaling . . . . . . . . . . . . . . . . . . . . . .
`1.5 Self-Affine Fractals . . . . . . . . . . . . . . . . . . . . . . . . . .
`1.6 More About Dimensionality . . . . . . . . . . . . . . . . . . . . .
`1.7 Percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
`
`2 Diffusion and Kinetics
`2.1 Random Walks and Regular Diffusion . . . . . . . . . . . . . . .
`2.2 Anomalous Diffusion . . . . . . . . . . . . . . . . . . . . . . . . .
`2.3 Fick’s Laws of Diffusion . . . . . . . . . . . . . . . . . . . . . . .
`2.4 Classical Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . .
`2.4.1 Passive Transport Processes . . . . . . . . . . . . . . . . .
`2.4.2 Reaction Processes: Diffusion- or Reaction-Limited? . . .
`2.4.3 Carrier-Mediated Transport . . . . . . . . . . . . . . . . .
`2.5 Fractal-like Kinetics
`. . . . . . . . . . . . . . . . . . . . . . . . .
`2.5.1
`Segregation of Reactants . . . . . . . . . . . . . . . . . . .
`2.5.2 Time-Dependent Rate Coefficients . . . . . . . . . . . . .
`2.5.3 Effective Rate Equations . . . . . . . . . . . . . . . . . . .
`2.5.4 Enzyme-Catalyzed Reactions . . . . . . . . . . . . . . . .
`2.5.5
`Importance of the Power-Law Expressions . . . . . . . . .
`2.6 Fractional Diffusion Equations
`. . . . . . . . . . . . . . . . . . .
`
`xi
`
`vii
`
`xvii
`
`1
`
`5
`6
`8
`9
`11
`11
`12
`12
`13
`14
`
`17
`18
`22
`23
`27
`28
`29
`30
`31
`31
`32
`34
`35
`36
`36
`
`Page 10
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`

`

`xii Contents
`
`3 Nonlinear Dynamics
`3.1 Dynamic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . .
`3.2 Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
`3.3 Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
`3.4 Sensitivity to Initial Conditions . . . . . . . . . . . . . . . . . . .
`3.5 Reconstruction of the Phase Space . . . . . . . . . . . . . . . . .
`3.6 Estimation and Control in Chaotic Systems . . . . . . . . . . . .
`3.7 Physiological Systems
`. . . . . . . . . . . . . . . . . . . . . . . .
`
`II MODELING IN BIOPHARMACEUTICS
`
`4 Drug Release
`4.1 The Higuchi Model . . . . . . . . . . . . . . . . . . . . . . . . . .
`4.2 Systems with Different Geometries . . . . . . . . . . . . . . . . .
`4.3 The Power-Law Model . . . . . . . . . . . . . . . . . . . . . . . .
`4.3.1 Higuchi Model vs. Power-Law Model . . . . . . . . . . . .
`4.4 Recent Mechanistic Models
`. . . . . . . . . . . . . . . . . . . . .
`4.5 Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . . . . .
`4.5.1 Verification of the Higuchi Law . . . . . . . . . . . . . . .
`4.5.2 Drug Release from Homogeneous Cylinders . . . . . . . .
`4.5.3 Release from Fractal Matrices . . . . . . . . . . . . . . . .
`4.6 Discernment of Drug Release Kinetics
`. . . . . . . . . . . . . . .
`4.7 Release from Bioerodible Microparticles . . . . . . . . . . . . . .
`4.8 Dynamic Aspects in Drug Release
`. . . . . . . . . . . . . . . . .
`
`39
`41
`42
`43
`45
`47
`49
`51
`
`53
`
`57
`58
`60
`63
`64
`67
`68
`69
`70
`75
`82
`83
`86
`
`89
`5 Drug Dissolution
`90
`. . . . . . . . . . . . . . . . . . . . .
`5.1 The Diffusion Layer Model
`92
`5.1.1 Alternative Classical Dissolution Relationships
`. . . . . .
`93
`5.1.2 Fractal Considerations in Drug Dissolution . . . . . . . .
`94
`5.1.3 On the Use of the Weibull Function in Dissolution . . . .
`97
`5.1.4
`Stochastic Considerations . . . . . . . . . . . . . . . . . .
`5.2 The Interfacial Barrier Model
`. . . . . . . . . . . . . . . . . . . . 100
`5.2.1 A Continuous Reaction-Limited Dissolution Model
`. . . . 100
`5.2.2 A Discrete Reaction-Limited Dissolution Model . . . . . . 101
`5.2.3 Modeling Supersaturated Dissolution Data . . . . . . . . 107
`5.3 Modeling Random Effects . . . . . . . . . . . . . . . . . . . . . . 109
`5.4 Homogeneity vs. Heterogeneity . . . . . . . . . . . . . . . . . . . 110
`5.5 Comparison of Dissolution Profiles . . . . . . . . . . . . . . . . . 111
`
`113
`6 Oral Drug Absorption
`6.1 Pseudoequilibrium Models . . . . . . . . . . . . . . . . . . . . . . 114
`6.1.1 The pH-Partition Hypothesis . . . . . . . . . . . . . . . . 114
`6.1.2 Absorption Potential . . . . . . . . . . . . . . . . . . . . . 115
`6.2 Mass Balance Approaches . . . . . . . . . . . . . . . . . . . . . . 117
`6.2.1 Macroscopic Approach . . . . . . . . . . . . . . . . . . . . 118
`
`Page 11
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`Contents
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` xiii
`
`6.2.2 Microscopic Approach . . . . . . . . . . . . . . . . . . . . 121
`6.3 Dynamic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
`6.3.1 Compartmental Models
`. . . . . . . . . . . . . . . . . . . 122
`6.3.2 Convection—Dispersion Models
`. . . . . . . . . . . . . . . 124
`6.4 Heterogeneous Approaches . . . . . . . . . . . . . . . . . . . . . . 129
`6.4.1 The Heterogeneous Character of GI Transit . . . . . . . . 129
`6.4.2
`Is in Vivo Drug Dissolution a Fractal Process? . . . . . . 130
`6.4.3 Fractal-like Kinetics in Gastrointestinal Absorption . . . . 132
`6.4.4 The Fractal Nature of Absorption Processes . . . . . . . . 134
`6.4.5 Modeling Drug Transit in the Intestines . . . . . . . . . . 136
`6.4.6 Probabilistic Model for Drug Absorption . . . . . . . . . . 142
`6.5 Absorption Models Based on Structure . . . . . . . . . . . . . . . 147
`6.6 Regulatory Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . 148
`6.6.1 Biopharmaceutics Classification of Drugs
`. . . . . . . . . 148
`6.6.2 The Problem with the Biowaivers . . . . . . . . . . . . . . 151
`6.7 Randomness and Chaotic Behavior . . . . . . . . . . . . . . . . . 158
`
`III MODELING IN PHARMACOKINETICS
`
`161
`
`165
`7 Empirical Models
`7.1 Power Functions and Heterogeneity . . . . . . . . . . . . . . . . . 167
`7.2 Heterogeneous Processes . . . . . . . . . . . . . . . . . . . . . . . 169
`7.2.1 Distribution, Blood Vessels Network . . . . . . . . . . . . 169
`7.2.2 Elimination, Liver Structure . . . . . . . . . . . . . . . . . 171
`7.3 Fractal Time and Fractal Processes . . . . . . . . . . . . . . . . . 174
`7.4 Modeling Heterogeneity . . . . . . . . . . . . . . . . . . . . . . . 175
`7.4.1 Fractal Concepts . . . . . . . . . . . . . . . . . . . . . . . 176
`7.4.2 Empirical Concepts
`. . . . . . . . . . . . . . . . . . . . . 177
`7.5 Heterogeneity and Time Dependence . . . . . . . . . . . . . . . . 178
`7.6 Simulation with Empirical Models
`. . . . . . . . . . . . . . . . . 181
`
`183
`8 Deterministic Compartmental Models
`8.1 Linear Compartmental Models
`. . . . . . . . . . . . . . . . . . . 184
`8.2 Routes of Administration . . . . . . . . . . . . . . . . . . . . . . 186
`8.3 Time—Concentration Profiles
`. . . . . . . . . . . . . . . . . . . . 187
`8.4 Random Fractional Flow Rates . . . . . . . . . . . . . . . . . . . 188
`8.5 Nonlinear Compartmental Models
`. . . . . . . . . . . . . . . . . 189
`8.5.1 The Enzymatic Reaction . . . . . . . . . . . . . . . . . . . 191
`8.6 Complex Deterministic Models
`. . . . . . . . . . . . . . . . . . . 193
`8.6.1 Geometric Considerations . . . . . . . . . . . . . . . . . . 194
`8.6.2 Tracer Washout Curve . . . . . . . . . . . . . . . . . . . . 195
`8.6.3 Model for the Circulatory System . . . . . . . . . . . . . . 197
`8.7 Compartmental Models and Heterogeneity . . . . . . . . . . . . . 199
`
`Page 12
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`xiv
`
` Contents
`
`205
`9 Stochastic Compartmental Models
`9.1 Probabilistic Transfer Models . . . . . . . . . . . . . . . . . . . . 206
`9.1.1 Definitions
`. . . . . . . . . . . . . . . . . . . . . . . . . . 206
`9.1.2 The Basic Steps
`. . . . . . . . . . . . . . . . . . . . . . . 208
`9.2 Retention-Time Distribution Models . . . . . . . . . . . . . . . . 210
`9.2.1 Probabilistic vs. Retention-Time Models . . . . . . . . . . 210
`9.2.2 Markov vs. Semi-Markov Models . . . . . . . . . . . . . . 212
`9.2.3
`Irreversible Models . . . . . . . . . . . . . . . . . . . . . . 214
`9.2.4 Reversible Models
`. . . . . . . . . . . . . . . . . . . . . . 217
`9.2.5 Time-Varying Hazard Rates . . . . . . . . . . . . . . . . . 222
`9.2.6 Pseudocompartment Techniques
`. . . . . . . . . . . . . . 225
`9.2.7 A Typical Two-Compartment Model . . . . . . . . . . . . 231
`9.3 Time—Concentration Profiles
`. . . . . . . . . . . . . . . . . . . . 235
`9.3.1 Routes of Administration . . . . . . . . . . . . . . . . . . 236
`9.3.2
`Some Typical Drug Administration Schemes . . . . . . . . 237
`9.3.3 Time-Amount Functions . . . . . . . . . . . . . . . . . . . 239
`9.3.4 Process Uncertainty or Stochastic Error . . . . . . . . . . 243
`9.3.5 Distribution of Particles and Process Uncertainty . . . . . 245
`9.3.6 Time Profiles of the Model
`. . . . . . . . . . . . . . . . . 249
`9.4 Random Hazard-Rate Models . . . . . . . . . . . . . . . . . . . . 251
`9.4.1 Probabilistic Models with Random Hazard Rates . . . . . 253
`9.4.2 Retention-Time Models with Random Hazard Rates . . . 258
`9.5 The Kolmogorov or Master Equations
`. . . . . . . . . . . . . . . 260
`9.5.1 Master Equation and Diffusion . . . . . . . . . . . . . . . 263
`9.5.2 Exact Solution in Matrix Form . . . . . . . . . . . . . . . 265
`9.5.3 Cumulant Generating Functions
`. . . . . . . . . . . . . . 265
`9.5.4
`Stochastic Simulation Algorithm . . . . . . . . . . . . . . 267
`9.5.5
`Simulation of Linear and Nonlinear Models . . . . . . . . 272
`9.6 Fractals and Stochastic Modeling . . . . . . . . . . . . . . . . . . 281
`9.7 Stochastic vs. Deterministic Models
`. . . . . . . . . . . . . . . . 285
`
`IV MODELING IN PHARMACODYNAMICS
`
`289
`
`293
`10 Classical Pharmacodynamics
`10.1 Occupancy Theory in Pharmacology . . . . . . . . . . . . . . . . 293
`10.2 Empirical Pharmacodynamic Models . . . . . . . . . . . . . . . . 295
`10.3 Pharmacokinetic-Dynamic Modeling . . . . . . . . . . . . . . . . 296
`10.3.1 Link Models . . . . . . . . . . . . . . . . . . . . . . . . . . 297
`10.3.2 Response Models . . . . . . . . . . . . . . . . . . . . . . . 303
`10.4 Other Pharmacodynamic Models . . . . . . . . . . . . . . . . . . 305
`10.4.1 The Receptor—Transducer Model
`. . . . . . . . . . . . . . 305
`10.4.2 Irreversible Models . . . . . . . . . . . . . . . . . . . . . . 305
`10.4.3 Time-Variant Models . . . . . . . . . . . . . . . . . . . . . 306
`10.4.4 Dynamic Nonlinear Models . . . . . . . . . . . . . . . . . 308
`10.5 Unification of Pharmacodynamic Models . . . . . . . . . . . . . . 309
`
`Page 13
`
`

`

`Contents
`
`xv
`
`10.6 The Population Approach . . . . . . . . . . . . . . . . . . . . . . 310
`10.6.1 Inter- and Intraindividual Variability . . . . . . . . . . . . 310
`10.6.2 Models and Software . . . . . . . . . . . . . . . . . . . . . 311
`10.6.3 Covariates . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
`10.6.4 Applications
`. . . . . . . . . . . . . . . . . . . . . . . . . 313
`
`315
`11 Nonclassical Pharmacodynamics
`11.1 Nonlinear Concepts in Pharmacodynamics . . . . . . . . . . . . . 316
`11.1.1 Negative Feedback . . . . . . . . . . . . . . . . . . . . . . 316
`11.1.2 Delayed Negative Feedback . . . . . . . . . . . . . . . . . 322
`11.2 Pharmacodynamic Applications . . . . . . . . . . . . . . . . . . . 334
`11.2.1 Drugs Affecting Endocrine Function . . . . . . . . . . . . 334
`11.2.2 Central Nervous System Drugs . . . . . . . . . . . . . . . 344
`11.2.3 Cardiovascular Drugs
`. . . . . . . . . . . . . . . . . . . . 348
`11.2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 350
`
`A Stability Analysis
`
`B Monte Carlo Simulations in Drug Release
`
`C Time-Varying Models
`
`353
`
`355
`
`359
`
`363
`D Probability
`. . . . . . . . . . . . . . . . . . . . . . . . . . . 363
`D.1 Basic Properties
`D.2 Expectation, Variance, and Covariance . . . . . . . . . . . . . . . 364
`D.3 Conditional Expectation and Variance . . . . . . . . . . . . . . . 365
`D.4 Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . 365
`
`E Convolution in Probability Theory
`
`F Laplace Transform
`
`G Estimation
`
`H Theorem on Continuous Functions
`
`I List of Symbols
`
`Bibliography
`
`Index
`
`367
`
`369
`
`371
`
`373
`
`375
`
`383
`
`433
`
`Page 14
`
`

`

`List of Figures
`
`1.1 The Koch curve . . . . . . . . . . . . . . . . . . . . . . . . . . . .
`1.2 The Sierpinski triangle and the Menger sponge . . . . . . . . . .
`1.3 Cover dimension . . . . . . . . . . . . . . . . . . . . . . . . . . .
`1.4 A 6 × 6 square lattice site model
`. . . . . . . . . . . . . . . . . .
`1.5 Percolation cluster derived from computer simulation . . . . . . .
`
`2.1 One-dimensional random walk . . . . . . . . . . . . . . . . . . . .
`2.2 Random walks in two dimensions . . . . . . . . . . . . . . . . . .
`2.3 Solute diffusion across a plane . . . . . . . . . . . . . . . . . . . .
`2.4 Concentration-distance profiles derived from Fick’s law . . . . . .
`2.5 Rate vs. solute concentration in Michaelis—Menten kinetics
`. . .
`
`3.1 Difference between random and chaotic processes . . . . . . . . .
`3.2 Schematic representation of various types of attractors . . . . . .
`3.3 The logistic map, for various values of the parameter θ . . . . . .
`3.4 The bifurcation diagram of the logistic map . . . . . . . . . . . .
`3.5 The Rössler strange attractor . . . . . . . . . . . . . . . . . . . .
`
`4.1 The spatial concentration profile of a drug . . . . . . . . . . . . .
`4.2 Case II drug transport with axial and radial release from
`a cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
`4.3 Fractional drug release vs. time . . . . . . . . . . . . . . . . . . .
`4.4 Schematic of a system used to study diffusion . . . . . . . . . . .
`4.5 Monte Carlo simulation of the release data . . . . . . . . . . . . .
`4.6 Number of particles inside a cylinder vs. time . . . . . . . . . . .
`4.7 Simulations with the Weibull and the power-law model . . . . . .
`4.8 Fluoresceine release data from HPMC matrices . . . . . . . . . .
`4.9 Buflomedil pyridoxal release from HPMC matrices . . . . . . . .
`4.10 Chlorpheniramine maleate release from HPMC
`K15M matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . .
`4.11 A percolation fractal embedded on a 2-dimensional
`square lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
`4.12 Plot of the release rate vs. time . . . . . . . . . . . . . . . . . . .
`4.13 Number of particles remaining in the percolation fractal
`. . . . .
`
`6
`7
`10
`14
`15
`
`19
`20
`24
`27
`30
`
`40
`42
`44
`46
`48
`
`59
`
`62
`65
`69
`70
`73
`74
`76
`77
`
`77
`
`79
`80
`81
`
`xvii
`
`Page 15
`
`

`

`xviii
`
` List of Figures
`
`4.14 Fitting of the power law to pseudodata . . . . . . . . . . . . . . .
`4.15 Triphasic drug release kinetics . . . . . . . . . . . . . . . . . . . .
`4.16 Conversion of pH oscillations to oscillations in drug flux . . . . .
`4.17 Schematic of pulsating drug delivery device . . . . . . . . . . . .
`
`84
`85
`86
`87
`
`90
`5.1 Basic steps in the drug dissolution mechanism . . . . . . . . . . .
`91
`5.2 Schematic representation of the dissolution mechanisms
`. . . . .
`95
`5.3 Accumulated fraction of drug dissolved vs. time . . . . . . . . . .
`98
`5.4 Cumulative dissolution profile vs. time . . . . . . . . . . . . . . .
`5.5 Plot of M DT vs. θ . . . . . . . . . . . . . . . . . . . . . . . . . .
`99
`5.6 Discrete, reaction-limited dissolution process
`. . . . . . . . . . . 102
`5.7 Dissolved fraction vs. generations (part I) . . . . . . . . . . . . . 103
`5.8 Dissolved fraction vs. generations (part II)
`. . . . . . . . . . . . 105
`5.9 Fraction of dose dissolved for danazol data
`(continuous model) . . . . . . . . . . . . . . . . . . . . . . . . . . 106
`5.10 Fraction of dose dissolved for danazol data
`(discrete model)
`. . . . . . . . . . . . . . . . . . . . . . . . . . . 106
`5.11 Fraction of dose dissolved for nifedipine data
`(discrete model)
`. . . . . . . . . . . . . . . . . . . . . . . . . . . 108
`
`6.1 Fraction of dose absorbed vs. Z . . . . . . . . . . . . . . . . . . . 117
`6.2 The small intestine as a homogeneous cylindrical tube . . . . . . 118
`6.3 Fraction of dose absorbed vs. the permeability . . . . . . . . . . 121
`6.4 Schematic of the ACAT model
`. . . . . . . . . . . . . . . . . . . 124
`6.5 Schematic of the velocity of the fluid inside the tube . . . . . . . 125
`6.6 Snapshots of normalized concentration inside the
`intestinal lumen . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
`6.7 A gastrointestinal dispersion model with
`spatial heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . 128
`6.8 Geometric representation of dissolution . . . . . . . . . . . . . . . 132
`6.9 Geometry of the heterogeneous tube . . . . . . . . . . . . . . . . 137
`6.10 Cross sections of the tube at random positions
`. . . . . . . . . . 138
`6.11 Mean transit times vs. the forward probability . . . . . . . . . . 141
`6.12 Frequency of mean transit times vs. time
`. . . . . . . . . . . . . 142
`6.13 Fraction of dose absorbed vs. An . . . . . . . . . . . . . . . . . . 146
`6.14 Three-dimensional graph of fraction dose absorbed . . . . . . . . 147
`6.15 The Biopharmaceutics Classification System (BCS).
`. . . . . . . 149
`6.16 Characterization of the classes of the QBCS . . . . . . . . . . . . 150
`6.17 The classification of 42 drugs in the plane of the QBCS . . . . . 152
`6.18 Dose vs. the dimensionless solubility—dose ratio . . . . . . . . . . 155
`6.19 Mean dissolution time in the intestine vs.
`effective permeability . . . . . . . . . . . . . . . . . . . . . . . . . 156
`6.20 Dose vs. 1/θ for the experimental data of Table 6.1 . . . . . . . . 157
`6.21 Phase plane