Thesis (MSc)--Stellenbosch University, 2011. / ENGLISH ABSTRACT: For solid tumours to grow and metastise, they need to pass through two distinct
stages: the avascular growth phase in which the tumour remains in a
limited diffusion size and the vascular growth phase where the invasion may
take place. In order to accomplish the transition from the former to the latter
growth phase, a solid tumour may secrete a substance known as tumour
angiogenesis factor (TAF) into the surrounding tissues to stimulate its own
blood vessels. Once the tumour has its own blood supply, it can invade other
parts of the body destroying healthy tissues organs by secreting the matrix
degrading enzymes (MDE). During the invasion, the adhesion both cell-cell
and cell-matrix play an extremely important role.
In this work, we review some mathematical models dealing with various stages
of development of solid tumours and the resulting reaction diffusion equations
are solved using the Crank-Nicolson finite differences scheme. We also present
a system of reaction-diffusion-taxis partial differential equations, with nonlocal
(integral) terms describing the interactions between cancer cells and the host
tissue. We then investigate the local and global existence of the solution of the
previous model using the semigroup method and Sobolev embeddings. / AFRIKAANSE OPSOMMING: Daar is twee afsonderlike fases nodig vir soliede kanker gewasse om te groei en
kwaadaardig te word: die avaskulêre groeifase waarin die gewas tot ’n sekere
diffusie grootte beperk word en die vaskulêre groei fase waar die indringing
plaasvind. Ten einde die oorgang tussen die twee fases te bewerkstellig, skei
die soliede gewas ân stof in die omliggende weefsel af wat bekend staan as
âtumor angiogenese factorâ (TAF). Dit stimuleer die vorming van die gewas se
eie bloedvate. Wanneer die gewas sy eie bloedtoevoer het, kan dit ander dele
van die liggaam indring en gesonde orgaanweefsel vernietig deur die afskeiding
van die âmatrix degrading enzymesâ (MDE). Gedurende hierdie proses speel
die sel-sel en sel-matriks interaksies ân belangrike rol. In hierdie werk het ons
ân paar wiskundige modelle vergelyk wat die verskillende stadiums van die ontwikkeling
van soliede gewasse beskryf. Die gevolglike diffusiereaksie vergelykings
is opgelos deur gebruik te maak van die âCrank-Nicolson finite differences
schemeâ. Ons bied ook ’n stelsel van âreaction-diffusion-taxisâ, met nie-lokale
(integrale) terme wat die interaksies tussen kankerselle en die gasheerweefsel beskryf. Ons stel dan ondersoek in na die lokale en globale bestaan van die
oplossing van die vorige model, met behulp van die semi-groep metode en die
Sobolev ingebeddings.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/18056 |
| Date | 12 1900 |
| Creators | Onana Eloundou, Jeanne Marie |
| Contributors | Banasiak, Jacek, Rewitzky, Ingrid, Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. |
| Publisher | Stellenbosch : Stellenbosch University |
| Source Sets | South African National ETD Portal |
| Language | en_ZA |
| Detected Language | Unknown |
| Type | Thesis |
| Format | 68 p. : ill. |
| Rights | Stellenbosch University |
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