Mathematical approaches to modelling and controlling blood thrombin formation [Elektronische Ressource] / vorgelegt von Sandra Maria da Cunha Órfão

Sujets

• Mathematics
• Gesundheit

Informations

• Publié par: rheinisch-westfalischen_technischen_hochschule_-rwth-_aachen
• Publié le: 01 janvier 2007
• Nombre de lectures: 60
• Langue: Deutsch
• Poids de l'ouvrage: 2 Mo

Extrait

Mathematical Approaches to Modelling
and Controlling Blood Thrombin
Formation
Von der Fakulta¨t fu¨r Mathematik, Informatik und Naturwissenschaften
der Rheinisch-Westfa¨lischen Technischen Hochschule Aachen
zur Erlangung des akademischen Grades einer Doktorin
der Naturwissenschaften genehmigte Dissertation
vorgelegt von
´Sandra Maria da Cunha Orfa˜o
aus Quelimane (Mosambik)
Berichter: Universita¨tsprofessor Dr. Gerhard Jank
Universita¨tsprofessor Dr. Sebastian Walcher
Professor Dr. Khosrhow Mottaghy
Tag der mu¨ndlichen Pru¨fung: 26.02.2007
DieseDissertationistaufdenInternetseitenderHochschulbibliothekonlineverfu¨gbar.Danksagung
AngefangenhatallesmitderWeihnachtsfeier2004desMasterstudiengangesBiome-
dical Engineering. Prof. Mottaghy kam zu Prof. Jank und mir und sagte, er habe
eine Idee, bei dem er hoffe, wir k¨onnen ihm helfen. Dass aus diesem Gespr¨ach
eineDissertationin Mathematikder Gro¨sseXL entstehen wu¨rde, warmir anfangs
nicht klar. Verantwortlich dafu¨r, dass daraus meine Arbeit entstanden ist, sind
Prof. Dr. Gerhard Jank, Prof. Dr. Khosrow Mottaghy und Prof. Dr. Sebastian
Walcher. Diesen drei Personen bin ich zu besonderem Dank verpflichtet.
Prof. Dr. G. Jank danke ich fu¨r seine st¨andige Unterstu¨tzung, nicht nur bei der
BetreuungmeinerArbeit,sondernauchfu¨ralles,waserfu¨rmichgetanhatseitich
in Aachen bin. Fu¨r mich ist er wie ein Mentor des Lebens und der Mathematik.
Prof. Dr. K. Mottaghy danke ich fu¨r die faszinierende Fragestellung und fu¨r die
st¨andige Unterstu¨tzung und Bereitschaft immer dann, wenn ich ihn gebraucht
habe.
Prof. Dr. S. Walcher ist sp¨ater eingestiegen. Ohne sein Interesse, seine Unterstu¨t-
zung und Bereitschaft ha¨tte meine Arbeit diese Tiefe in so kurzer Zeit nicht erre-
icht. Vielen Dank!
Mein perso¨nlicher Dank gilt auch Gehrt Hartjen und Daniel Robertz fu¨r die
MAPLEBeratungundDirkMeierlingbeiHilfestellungeninGraphentheorie.Nicht
zu vergessen Wolfgang Kromen, Victor Schneider, Filipe Miranda und Miguel
Alonso: Muito obrigada por terem lido e comentado o meu trabalho! Konstantin
Weiß danke ich sowohl fu¨r die Unterstu¨tzung bei graphischen Darstellungen als
auch fu¨r die Hilfe beim Poster, das ich in Bologna pr¨asentiert habe. Fu¨r die un-
bezahlbare Unterstu¨tzung in allen Lagen des Lebens sage ich Dir Konstantin:
Spassiba!
An meine restlichen Freunde, die mehr oder weniger von u¨berall her kommen
und die u¨berall wohnen, die aber meine Odyssee immer begleitet haben, sage ich
einfach: Obrigada! Gracias! Merci! Thanks! Danke!
AnFrauVolkmann,dieProfessoren,ArbeitskollegenundFreundedesLehrstuhlII
fu¨r Mathematik sage ich Danke fu¨r die scho¨ne Zeit mit Euch/Ihnen, insbesonderefu¨r die angenehme Atmospha¨re, und die netten Feierabende, wo ich immer gerne
mitgemacht habe.
Aosmeusirma˜oseaosrestantesfamiliaresagradec¸oa atenc¸a˜o,ocarinhoeoapoio
que me deram ao longo destes anos.
Bem hajam.
´Aachen, Januar 2007 Sandra Orfa˜oPreface
The title of this thesis reveals the interdisciplinary nature of my investigation,
where different mathematical methods were used to handle the questions of mod-
elling,analyzingandcontrollingthrombinformationinbloodcoagulationsystems.
Different scientific disciplines have different languages and different ways to build
structures and processes, so that the flow between knowledge and new findings is
usually restricted to one particular field. As a consequence, disciplines of exact,
empirical or experimental nature can hardly communicate with each other. On
theother hand, systems likethe blood coagulationsystem, cannotbe explainedor
understood only by using the laws or axioms of a single science. Thus, one of the
aims of this thesis was to create a science to science interface and a common plat-
formofknowledgebasedindifferentmathematicalapproaches.Everythingstarted
thereforebycollectingthenecessaryinformationfromMathematics,Medicineand
Biochemistry. So, there must be a proper way of stating the right questions and
getting the right answers. Thereby, one of the essential points was to pick up a
simple but by no means simplistic form to communicate contents.
The process of collecting relevant information from scientists with different back-
grounds is not a simple matter. However, the process of filtering and synthesizing
information from foreign disciplines is in general more demanding. In this phase,
the first task is to select information by rhetoricasking: what is really important?
what do I really need from the foreign disciplines? shall I go deeper into some of
these subjects? is there some piece missing? where do I get it? The second task
is to translate the information and state the problem in mathematical terms. The
later here, one has to decide which mathematical branch or branches are more
suitable to built the model and make further analysis. Thereby, it is crucial to
investigate whether there are known approaches in the literature and to keep in
mindthattherearesomenaturalrelevantparameterswithaveryspecificmeaning
that cannot be changed. So, the model has to reflect at least some of the most
importantcharacteristicsof the system. Whenever possible, it should providenew
insights about the mechanism that is the object of study. Last but not least, the
results of the investigation should be made intelligible to all potential differentreaders with variable scientific backgrounds. This is of major importance for the
differentscientiststo usetheinformationgainedandtoprogressintheirownarea.
Inparticular,mathematicians experiencewhether the approachcanbe fullybased
on classical results or whether there is still the need of new results.
Thrombin is the essential enzyme product of the blood coagulation process. Since
theearlyinvestigationsonthisfield,thebloodcoagulationprocesshasbeenrepre-
sented as a cascade of enzymatic reactions. Nowadays it is known to respond in a
thresholdmanner,involvingnumerous intermeshedcontrolsincluding feedforward
and feedbackward loops. The regulation of the production of thrombin is vital
to the maintenance of the hemostatic balance in humans. However, uncontrolled
generation of this enzyme can lead to physiological disaster. From the medical
literature it is known that foreign surface contact, e. g. during application of arti-
ficialorgans,isastrongactivatorofwholeofthesystem.Thus,intelligibilityofthe
roles of the system components in this regulation is important for a therapeutic
control of thrombotic and bleeding disorders. Chapter 1 contains the physiolog-
ical background, where the principal procoagulatory and anticoagulatory factors
and the two pathways leading to the formation of a fibrin clot are introduced.
Furthermore, some aspects concerning the kinetics of coagulation are discussed
together with a brief description of the contribution of mathematical modelling to
the understanding of the dynamics inherent to this physiological system.
On the one hand, it is important for instance to know how biochemists deduce
from a reaction scheme the differential equations they work with. On the other
hand,itisalsoimportanttopresentcontentsinaterminologyandformalismmore
standard in control and in dynamical systems theories. Thus, we put together the
forthcoming information from the different literature sources in such a way that
references in subsequent chapters areeasily made. Terms like stoichiometry or de-
ficiency ofanetworkariseandhelptogaininsightsintothestructureofachemical
network and to understand how they influence its dynamics. All this is done in
the first part of Chapter 2. Furthermore, since the blood coagulation system in-
volves a series of enzymatic reactions, the second part of this chapter explains
kinetic aspects of enzyme-catalyzed reactions. In particular, concepts like quasi-
steady assumption, Michaelis-Menten equation or enzyme inhibition are briefly
introduced.
Several attempts of modelling a part of the blood coagulation system include a
stiff system of nonlinear differential equations with unknown parameters. These
parameters are the reaction constants and the initial concentration of the coag-
ulation factors, which are normally estimated by fitting experimental data. The
number of parameters and equations is however considerably large, what might
result in low parameter sensitivity. Moreover, the reaction mechanism is also not
precisely known. In fact, there might be reactions that have been shown to ex-
ist in principle but do not occur in reality [SHH97]. Although kinetic analysis ofthe individual reactions have been reported in the literature, an analysis of the
complete reaction network has been given less attention. Altogether, another aim
for this thesis was to make a careful mathematical analysis of two of the most
cited models among the scientific community investigating the mechanisms of the
blood coagulation cascade. One model is due to Stortelder, Hemker and Hemker
[SHH97] and the other one is due to Jones and Mann [JoMa94].
These models are presented in Chapter 3. With the tools provided in Chapter
2, an analysis of the stoichiometry of both models is made and the numerical
solution provided. Besides stoichiometry, we interpreted the reaction scheme of
Jones