Analysis and Implementation of an Age Structured Model of the Cell Cycle
(2017) In Master's Theses in Mathematical Sciences FMA820 20171Mathematics (Faculty of Engineering)
 Abstract
 In an agestructured model originating from cancer research, the cell cycle is divided into two phases: Phase 1 of variable length, consisting of the biologically so called G1 phase, and Phase 2 of fixed length, consisting of the so called S, G2 and M phases. A system of nonlinear PDEs along with initial and boundary data describes the number densities of cells in the two phases, depending on time and age (where age is the time spent in a phase). It has been shown that the initial and boundary value problem can be rewritten as a coupled system of integral equations, which in this M.Sc. thesis is implemented in Matlab using the trapezoidal and Simpson rule. In the special case where the cells are allowed to grow without restrictions, the... (More)
 In an agestructured model originating from cancer research, the cell cycle is divided into two phases: Phase 1 of variable length, consisting of the biologically so called G1 phase, and Phase 2 of fixed length, consisting of the so called S, G2 and M phases. A system of nonlinear PDEs along with initial and boundary data describes the number densities of cells in the two phases, depending on time and age (where age is the time spent in a phase). It has been shown that the initial and boundary value problem can be rewritten as a coupled system of integral equations, which in this M.Sc. thesis is implemented in Matlab using the trapezoidal and Simpson rule. In the special case where the cells are allowed to grow without restrictions, the system is uncoupled and possible to study analytically, whereas otherwise, a nonlinearity has to be solved in every step of the iterative equation solving. The qualitative behaviour is investigated numerically and analytically for a wide range of model components. This includes investigations of the notions of crowding, i.e. that cell division is restricted for large population sizes, and quorum sensing, i.e. that a small enough tumour can eliminate itself through cell signalling. In simulations, we also study under what conditions an almost eliminated tumour relapses after completed therapy. Moreover, upper bounds for the number of dividing cells at the end of Phase 2 at time t are determined for specific cases, where the bounds are found to depend on the existence of so called cancer stem cells. Lastly, a careful error analysis of the Matlab implementation is performed both in a linear and in a nonlinear case. (Less)
 Popular Abstract (Swedish)
 I en åldersbaserad modell av cellcykeln som härstammar från cancerforskning kan cellcykeln delas in i två faser: en av variabel längd och en av fix längd. Det totala antalet celler i en tumör kan, tillsammans med antalet celler som just avslutat en cykel och ska genomgå celldelning, beskrivas genom ett system av integralekvationer. I arbetet studeras detta system dels analytiskt och dels numeriskt, genom förfinad implementering av modellen i Matlab och simulering för många olika val av modellens ingående komponenter. Bland annat undersöks huruvida återväxt av en tumör efter avslutad behandling beror av så kallade cancerstamceller.
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/studentpapers/record/8925647
 author
 Broms, Anna ^{LU}
 supervisor

 Sara Maad Sasane ^{LU}
 Gustaf Söderlind ^{LU}
 organization
 alternative title
 Åldersbaserad modell av cellcykeln: Analys och implementering
 course
 FMA820 20171
 year
 2017
 type
 H2  Master's Degree (Two Years)
 subject
 keywords
 age structured model, cell cycle, cell population dynamics, nonlinear integral equations, Romberg's method
 publication/series
 Master's Theses in Mathematical Sciences
 report number
 LUTFMA33272017
 ISSN
 14046342
 other publication id
 2017:E51
 language
 English
 id
 8925647
 date added to LUP
 20170928 14:41:21
 date last changed
 20170928 14:41:21
@misc{8925647, abstract = {In an agestructured model originating from cancer research, the cell cycle is divided into two phases: Phase 1 of variable length, consisting of the biologically so called G1 phase, and Phase 2 of fixed length, consisting of the so called S, G2 and M phases. A system of nonlinear PDEs along with initial and boundary data describes the number densities of cells in the two phases, depending on time and age (where age is the time spent in a phase). It has been shown that the initial and boundary value problem can be rewritten as a coupled system of integral equations, which in this M.Sc. thesis is implemented in Matlab using the trapezoidal and Simpson rule. In the special case where the cells are allowed to grow without restrictions, the system is uncoupled and possible to study analytically, whereas otherwise, a nonlinearity has to be solved in every step of the iterative equation solving. The qualitative behaviour is investigated numerically and analytically for a wide range of model components. This includes investigations of the notions of crowding, i.e. that cell division is restricted for large population sizes, and quorum sensing, i.e. that a small enough tumour can eliminate itself through cell signalling. In simulations, we also study under what conditions an almost eliminated tumour relapses after completed therapy. Moreover, upper bounds for the number of dividing cells at the end of Phase 2 at time t are determined for specific cases, where the bounds are found to depend on the existence of so called cancer stem cells. Lastly, a careful error analysis of the Matlab implementation is performed both in a linear and in a nonlinear case.}, author = {Broms, Anna}, issn = {14046342}, keyword = {age structured model,cell cycle,cell population dynamics,nonlinear integral equations,Romberg's method}, language = {eng}, note = {Student Paper}, series = {Master's Theses in Mathematical Sciences}, title = {Analysis and Implementation of an Age Structured Model of the Cell Cycle}, year = {2017}, }