Modeling in Biopharmaceutics, Pharmacokinetics and Pharmacodynamics: Homogeneous and Heterogeneous Approaches

497 Pages·2016·9.46 MB·English

by  Panos Macheras

Interdisciplinary Applied Mathematics 30
Panos Macheras
Athanassios Iliadis
Modeling in
Biopharmaceutics,
Pharmacokinetics and
Pharmacodynamics
Homogeneous and Heterogeneous
Approaches
Second Edition
Modeling in Biopharmaceutics, Pharmacokinetics
and Pharmacodynamics
Interdisciplinary Applied Mathematics
Volume30
Editors
S.S.Antman P.Holmes
L.Greengard
SeriesAdvisors
LeonGlass RobertKohn
P.S.Krishnaprasad JamesD.Murray
ShankarSastry JamesSneyd
Problems in engineering, computational science, and the physical and biological
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Panos Macheras • Athanassios Iliadis
Modeling in
Biopharmaceutics,
Pharmacokinetics and
Pharmacodynamics
Homogeneous and Heterogeneous
Approaches
Second Edition
123
PanosMacheras AthanassiosIliadis
FacultyofPharmacy FacultyofPharmacy
NationalandKapodistrianUniversity Aix-MarseilleUniversity
ofAthens Marseille,France
Athens,Greece
ISSN0939-6047 ISSN2196-9973 (electronic)
InterdisciplinaryAppliedMathematics
ISBN978-3-319-27596-3 ISBN978-3-319-27598-7 (eBook)
DOI10.1007/978-3-319-27598-7
LibraryofCongressControlNumber:2015957958
MathematicsSubjectClassification(2010):92C45,62P10,74H65,60K20
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PanosMacherasandAthanassiosIliadis

Preface to the First Edition
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Greatartisfoundwherevermanachievesanunderstandingofselfandisabletoexpress
himselffullyinthesimplestmanner.
OdysseasElytis(1911–1996)
1979NobelLaureateinLiterature
TheMagicofPapadiamantis
Biopharmaceutics, pharmacokinetics, and pharmacodynamics are the most
importantpartsofpharmaceuticalsciencesbecausetheybridgethegapbetweenthe
basicsciencesandtheclinicalapplicationofdrugs.Themodelingapproachesinall
threedisciplinesattemptto:
• Describe the functional relationships among the variables of the system under
study.
• Provideadequateinformationfortheunderlyingmechanisms.
Duetothecomplexityofthebiopharmaceutic,pharmacokinetic,andpharmaco-
dynamicphenomena,novelphysicallyphysiologicallybasedmodelingapproaches
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 disso-
lution 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 realized 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,
weborrowedgeometric,diffusional,anddynamicconceptsandtoolsfromphysics
and mathematics and applied them to the analysis of complex biopharmaceutic,
pharmacokinetic, and pharmacodynamic phenomena. Thus, this book grew out of
vii
viii PrefacetotheFirstEdition
our conversations with fellow colleagues, 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,andilluminatewiththeiruseoftheintrinsiccomplexityofdrugprocesses
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,pharmacokinetics,andpharmacodynamics.
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 models
usedindrugdissolution,release,andabsorption;PartIII,Chapters7–11,presenting
empirical, compartmental, and stochastic pharmacokinetic models; and Part IV,
Chapters12and13,presentingclassicalandnonclassicalpharmacodynamicmod-
els. The level of mathematics required for understanding each chapter varies.
Chapters 1 and 2 require undergraduate-level algebra and calculus. Chapters 3–
8, 12, and 13 require knowledge of upper undergraduate- to graduate-level linear
analysis,calculus,differentialequations,andstatistics.Chapter11requiresknowl-
edgeofprobabilitytheory.
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
ofdiffusionandkineticsapply.Thetermheterogeneousisusedforprocessestaking
placeindisorderedmediaorundertopologicalconstraintswhereclassicaldiffusion-
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
dynamicsasdelineatedinChapter3.
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
imageorabstractionofreality:amental,physical,ormathematicalrepresentationor
descriptionofanactualprocess,suitableforacertainpurpose.Themodelneednot
beatrueandaccuratedescriptionoftheprocess,norneedtheuserhavetobelieveso,
inordertoserveitspurpose.Herein,onlymathematicalmodelsareused.Eitherpro-
cessesormodelscanbeconceivedasboxesreceivinginputsandproducingoutputs.
The boxes may be characterized as gray or black, when the internal mechanisms
and parameters are associated or not with a physical interpretation, respectively.
Thesystemisacomplexentityformedofmany,oftendiverse,interrelatedelements
servingacommongoal.Alltheseelementsareconsideredasdynamicprocessesand
models.Here,deterministic,random,orchaoticrealprocessesandthemathematical
modelsdescribingthemwillbereferencedassystems.Whenevertheword“system”
hasaspecificmeaninglikeprocessormodel,itwillbeaddressedassuch.
Forcertainprocesses,itisappropriatetodescribegloballytheirpropertiesusing
numerical techniques that extract the basic information from measured data. In
PrefacetotheFirstEdition ix
the domain of linear processes, such techniques are correlation analysis, spectral
analysis,etc.andinthedomain ofnonlinear processes,thecorrelationdimension,
the Lyapunov exponent, etc. These techniques are usually called nonparametric
modelsor,simply,indices.Formoreadvancedapplications,itmaybenecessaryto
usemodelsthatdescribethefunctionalrelationshipsamongthesystemvariablesin
terms of mathematical expressions like difference or differential equations. These
models assume a prespecified parameterized structure. Such models are called
parametricmodels.
Usually, a mathematical model simulates a process behavior, in what can be
termedaforwardproblem.Theinverseproblemis,giventheexperimentalmeasure-
mentsofbehavior,whatisthestructure?Adifficultproblem,butanimportantone
forthesciences.Theinverseproblemmaybepartitionedintothefollowingstages:
hypothesis formulation, i.e., model specification, definition of the experiments,
identifiability,parameterestimation,experiment,andanalysisandmodelchecking.
Typically, from measured data, nonparametric indices are evaluated in order to
revealthebasicfeaturesandmechanismsoftheunderlyingprocesses.Then,based
onthisinformation,severalstructuresareassayedforcandidateparametricmodels.
Nevertheless,inthisbookwelookonlyintovariousaspectsoftheforwardproblem:
giventhestructureandtheparametervalues,howdoesthesystembehave?
Here,theuseoftheterm“model”followsKac’sremark,“modelsarecaricatures
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
sameprocess.Also,Fourierremarked,“natureisindifferenttowardthedifficulties
it causes a mathematician”; in other words the mathematics should be dictated
by the biology and not vice versa. For choosing among such competing 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
purposesofitsuse.So,forthesameprocess,onemaydevelopmodelsforprocess
identification,simulation,control,etc.Inthisvein,thetouristmapofAthensandthe
system controlling the urban traffic in Marseille are both tools associated with the
reallifeinthesecities.Thefirstisanidentificationmodelandthesecond,acontrol
model.
Over the years we have benefited enormously from discussions and collabora-
tionswithstudentsandcolleagues.InparticularwethankP.Argyrakis,D.Barbolosi,
A. Dokoumetzidis, A. Kalampokis, V. Karalis, K. Kosmidis, C. Meille, E. Rinaki,
andG.Valsami.WewishtothankJ.Lukaswhosesuggestionsandcriticismsgreatly
improvedthemanuscript.
Piraeus,Greece P.Macheras
Marseille,France A.Iliadis
August2005