Splines and compartment models : an introduction

349 Pages·2013·2.893 MB·English

by  Karl-Ernst Biebler, Michael Wodny

SPLINES AND
COMPARTMENT
MODELS
An Introduction
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SPLINES AND
COMPARTMENT
MODELS
An Introduction
Karl-Ernst Biebler
Michael Wodny
Ernst Moritz Arndt University of Greifswald, Germany
World Scientific
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SPLINES AND COMPARTMENT MODELS
An Introduction
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Preface
Theapplicationofmathematicsinlifesciencesfirstrequirestheformulation
of adequate models of biological processes that allow the quantitative eval-
uation of life processes by means of observations and experiments. With
regardtothis, theknowledgewithreferencetotheobservedbiologicalpro-
cesses, the preconditions and characteristics of the applied mathematical
models as well as the conditions surrounding data collection, need to be
taken into account. In this entire context it is effective to develop spe-
cific quantitative methods for the evaluation of data and to characterize
attributes mathematically, thereby justifying conclusions and interpreting
results comprehensively. This synopsis of problems is a characteristic of
biometric work, which due to its formulation is interdisciplinary. Typical
questions continue to be brought to light, for example:
• Can the conditions required by the mathematical method be seen
as fulfilled in the observed examples?
• Doesasolutionexistforthemathematicalproblemassociatedwith
the general problem definition?
• Is there exactly one such solution?
• Can it be proven that this solution possess desirable characteris-
tics?
• Aretheevaluationmodelandpossibilitiesfordataacquisitioncon-
sistent?
• Do numerical problems arise when applying processes?
Under such general points of views, two themes are discussed in this book:
splinesandcompartmentmodels. Theirapplicationcanbeseenindifferent
areas of the sciences and technology.
Whydoesonedealwithsuchdifferentmathematicalconceptsinthisbook?
v
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vi Splines and Compartment Models - An Introduction
It is common for them that they both can be used for the modeling of
courses. Spline functions result from a general approach. Watched courses
canbereproducedverywellunderreferencetocertainoptimizationcriteria.
Splinesaredeterminedbyarelativelyhighnumberofparameters. However,
these are often not well interpretable according to a specific question.
Compartmentmodelsrepresentadescriptionofspecificprocessesbymeans
of differential equations. Their parameters characterize the process. The
number of model parameters is relatively small here. The data is generally
not as precisely reproducible as by means of splines. The demand to judge
the quality of the goodness of fit consequently arises. To do so, residence
time distributions are taken to a connection with compartment models
of pharmacokinetics. This way the model parameters can be calculated
statistically from the measured drug concentration data. We propose the
variedMinimum-χ2 estimation. Splinesareusedforit. Goodnessoffitcan
now be checked with a statistical test. This procedure is new in the field
of pharmacokinetics.
Examples are given in the context of life sciences with the appropriate
typicalterminology. Atthesametime,mathematicaltermsareneededthat
cannot be explained in detail here. The reader would find it useful to be
familiarwithbasicknowledgeofanalysis,algebra,statisticsandprobability
calculus as well as the theory of ordinary differential equations. Detailed
knowledge of these areas however is not assumed.
In this book we took care to include the history of the presented ideas and
include references with regard to this. The historical comparison is not to
be seen as just a reference to the scientists of the past. It helps to enforce
therelativizationoftheownwork,motivatesstudentsandfurtherallowthe
readertoresearchthesourcesthemselves. Tobegin,narrowinginonsimple
models seems to be advisable for the solid application of mathematical
models in the life sciences. This simplifies the detailed clarification of their
conditions of application, reduces the demand on the extent of observation
and in many cases satisfies the purpose.
The literature listed in the references confines itself to the titles quoted
in the text. Numerous additional publications which were evaluated but
did not explicitly contribute to results are not listed. This concerns stan-
dardpharmacologyorpharmacokineticstextbooks,publicationsconcerning
computerprograms,applicationsofpharmacokineticalmethods(inclusively
aboutmethodsofcalculation)forthecharacterizationofcertainpharmaka,
literatureaboutstochasticsandtheirapplications, contributionsaboutnu-
meric methods, etc.
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Preface vii
Different spline functions are dealt with in Part I. The starting point is
the task of describing an unknown functional dependency y = f(x). In
the life sciences this is often the change of quantity over time. The typical
situation with regard to the data is the observation of a finite number of
y-values that are understood as the function values of the respective inde-
pendent x-values. A number of m value pairs (x ,y ), i = 1,2,...,m, are
i i
therefore given. Sometimes subject specific examples allow for the defini-
tion of a parametric function class from which f(x) stems. With this it is
then possible, for example with the method of least squares, to calculate
the function parameters from the data. It is often the case though that no
special knowledge exists about f(x). This is where spline functions can be
used. Thenaturalsplinefunctionspossessaspecialsmoothingcharacteris-
tic. Itcanbasicallybeassumedthatthecourseoff(x)ismostlydescribed
by the value pairs (x ,y ), i =1,2,...,m. Unobserved oscillations are not
i i
hidden between individual points. This along with the numerically stable
calculability of splines lead to their wide use. To begin, general natural
splinesareintroduced. Followingthis,cubicandquadraticsplinefunctions
are described in more detail. In each case, the interpolation task is first
formulated,andthentheirsolutionisdevelopedanddiscussed. Thismakes
senseandisnecessarybecausetheconstructionofsmoothingsplinesisdone
in two steps. In the first step the still unknown values s(x ) of the spline
i
function are calculated at x . In the second step, the uniquely determined
i
interpolating spline is constructed for these. A certain optimization prob-
lem corresponds with every smoothing spline. This problem determines
the main characteristics of the solution function. The individual problems
are always first approached with regard to their unique solvability. If this
unique solution is not given, then the problem is modified appropriately.
In each case, a great emphasis is put on the numeric determination of
the solution. The formal representation of each solution is given in detail.
This makes the reconstruction of the problem possible with, for example,
Mathematicafi, Maplefi or the SASfi Procedure IML.
Splinefunctionsbuildalinearfunctionspace. Thismakestheconstruction
of an average function that is representative of a certain group of objects
or individuals possible. From the point of view of the user, this is often
an advantage of the spline approach over other nonlinear models. The
corresponding examples are related to the medical field. Suggestions to
determine reference regions for clinical parameters are made.
Part II encompasses compartment models. The application of these mod-
els can be found in pharmacokinetics, physiology and in clinical medicine.
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viii Splines and Compartment Models - An Introduction
On hand of an individually observed course of concentration over time,
so-called individual kinetics are to be described quantitatively. Mathemat-
ically speaking, this deals mainly with systems of linear ordinary differen-
tial equations with constant coefficients. Variations of this deterministic
approach are shortly brought to light. For example, if a conceivable delay
of effect is to be observed in the model, then differential equations with
retarded argument come into question. Even for the simplest case it re-
sultsthatthesolutionofsuchdifferentialequationsareoscillatingfunctions
with negative values. Such models are therefore of no interest for the field
ofpharmacokinetics. TheTwo-compartment-ivmodel,otherwiseknownas
thehomogeneoussystemsoflinearordinarydifferentialequationswithcon-
stant coefficients, is the subject of additional observations. The solution to
systems of linear ordinary differential equations with constant coefficients
iswellknown. Soistheexistenceanduniquenessofthissolution. Tostart,
the 16 possible variants of these Two-compartment-iv models and their re-
spective solutions to the differential system of equations are given. On one
hand, data regarding individual kinetics are obtained and a function from
a certain function class can be fit to it. The parameters of the function
that are obtained are called system parameters. On the other hand there
are also the model parameters of the Two-compartment-iv model that also
define the solution function. With regard to the selection of a model it
is explored which model allows its parameters to be calculated from the
system parameters and which model can even be identified on hand of the
system parameters. With regard to this it needs to be seen which com-
partment can be observed. The calculation of the system parameters from
availabledatacanbedoneindifferentwaysandwithdifferentresults. The
amazing amount of practical methods and computer programs is shown in
an overview of literature.
Dothedataandmodelevenfittogether? Ideallytheselectionofmodelsand
the calculation of parameters are handled together. A statistical process
is developed for this: statistical parameter estimation and test of goodness
of fit. A stochastic model is needed for this. The random variable is the
residencetimeofapharmaconmoleculeinanorganism. Probabilitydensi-
ties can be derived from solutions to differential equations. It is examined
which of the Two-compartment-iv models correspond with residence time
distributions For the given data, the classic Pearson test can determine
the appropriateness of the model. Since the random variable can only be
observedindirectly, theasymptoticallyequivalentvariedMinimum-χ2 esti-
mationoccursinplaceofthemaximum-likelihoodestimationoftheparam-
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Preface ix
eters. Thetheoryofestimationallowsforthespecificationofcharacteristics
of the process for calculating parameters. Time restricted observation can
be taken into account by model transition to truncated distributions. An
application of the two compartment iv model that is of interest to clinical
medicine is finally handled: the method by Dost for the quantitative de-
scriptionofthemultipleapplicationofmedicationisdetailedandextended.
In Part III the reader will find Mathematicafi programs for selected prob-
lems. Theymaysupporttheapplicationofmethodsdevelopedinthisbook.
These programs are written straightforward under use of the terminology
of the employed formulas.
Thebookturnstolifescientists,theoreticalmedicalpractitionersandmath-
ematicians engaged in life sciences as well as students of these disciplines.
We use this material in our lectures on biometry for students studying
biomathematics, human biology and medicine.
We would like to thank our colleagues of the Medical Faculty of the
Ernst-Moritz-Arndt-University in Greifswald for the helpful discussions
around medical questions, students in our biometry courses for their in-
terest in the topic, Paul Rosenthal, M.Sc.(Mathematics), Kathrin Wu¨nsch
M.Sc.(Physics) and Carolin Malsch, B.Sc.(Biomathematics) for her tech-
nical help with the manuscript and her suggestions for corrections as well
as Frieda Kaiser, M.Sc. (Medical Informatics), Victoria, B.C., Canada, for
her dedicated work in translating the manuscript into English.
Greifswald, Germany Karl-Ernst Biebler
2013 Michael Wodny