| Abstract |
Mathematical models, that depict the dynamics of a cancer cell population
growing out of the human body (in vitro) in unconstrained microenvironment
conditions, are considered in this thesis. Cancer cells in vitro grow and
divide much faster than cancer cells in the human body, therefore, the
effects of various cancer treatments applied to them can be identified
much faster. These cell populations, when not exposed to any cancer
treatment, exhibit exponential growth that we refer to as the balanced
exponential growth (BEG) state. This observation has led to several
effective methods of estimating parameters that thereafter are not
required to be determined experimentally.
We present derivation of the age-structured model and its theoretical
analysis of the existence of the solution. Furthermore, we have obtained
the condition for BEG existence using the Perron-Frobenius theorem. A
mathematical description of the cell-cycle control is shown for
one-compartment and two-compartment populations, where a compartment refers
to a cell population consisting of cells that exhibit similar kinetic
properties. We have incorporated into our mathematical model the required
growing/aging times in each phase of the cell cycle for the biological
viability. Moreover, we have derived analytical formulae for vital
parameters in cancer research, such as population doubling time, the
average cell-cycle age, and the average removal age from all phases, which
we argue is the average cell-cycle time of the population. An estimate of
the average cell-cycle time is of a particular interest for biologists and
clinicians, and for patient survival prognoses as it is considered that
short cell-cycle times correlate with poor survival prognoses for patients.
Applications of our mathematical model to experimental data have been
shown. First, we have derived algebraic expressions to determine the
population doubling time from single experimental observation as an
alternative to empirically constructed growth curve. This result is
applicable to various types of cancer cell lines. One option to extend
this model would be to derive the cell cycle time from a single
experimental measurement. Second, we have applied our mathematical model
to interpret and derive dynamic-depicting parameters of five melanoma cell
lines exposed to radiotherapy. The mathematical result suggests there are
shortcomings in the experimental methods and provides an insight into the
cancer cell population dynamics during post radiotherapy. Finally, a
mathematical model depicting a theoretical cancer cell population that
comprises two sub-populations with different kinetic properties is
presented to describe the transition of a primary culture to a cell line
cell population.
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