Practical approaches to the application of mathematical modelling in oncology: from model investigation to data-driven model construction
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Date
2017-07
Authors
Bolton, Larisse
Journal Title
Journal ISSN
Volume Title
Publisher
University of the Free State
Abstract
Two different practical approaches to the application of mathematical modelling in oncology are
investigated. The first approach involved a comparative model investigation, where the focus
was on determining whether a fractional-order Gompertz model or the ordinary integer-order
Gompertz model would be the best descriptor of growth for a particular set of tumour growth
volumes. Herein, fractional calculus is used with the ordinary Gompertz model to propose a
fractional-order version of the model, with orders between 0 and 1. After obtaining the solution of
the fractional-order model through the application of fractional calculus, calibration of the solution
together with the solution of the ordinary Gompertz model was carried out with the use of a set
of Rhabdomyosarcoma volumes in a mouse. This generated the approximate fractional-order for
the proposed model that would best describe the data. Using the approximate fractional-order
obtained for the proposed model during calibration, as well as the ordinary Gompertz model, a
further comparative investigation was conducted into the ability of both models to predict a data
point of the dataset not included in the calibration process. At conclusion of these investigations
it was found that the fractional-order Gompertz model of order 0.68 could capture the growth
behaviour represented in the data better than that of the ordinary Gompertz model. The proposed
model not only fit the data better, but also had better prediction capability over the period of
observation versus its integer-order counterpart.
The second approach involved a data-driven investigation leading to the construction of a mathe-
matical model of acute lymphoblastic leukaemia. The aim of this investigation was to determine
whether a model of reasonable suitability could be proposed and whether such a model could gen-
erate meaningful and relevant insights not only into the system under investigation, but also into
the model. The latter would form the framework for further investigation to generate a tool with
not only explanatory but also predictive capability for clinicians. More specifically, clinical and
mathematical literature was used to propose a mathematical model considering lymphoblasts as
well as the granulocytic and monocytic population for patients diagnosed with variants of B-cell
acute lymphoblastic leukaemia during induction chemotherapy. The proposed model, consisting of
a system of delay differential equations, was calibrated and tested using retrospective data of paedi-
atric patients diagnosed with a variety of B-cell acute lymphoblastic leukaemia who had undergone
induction chemotherapy at Universitas Academic Hospital in Bloemfontein, South Africa. Therein,
the model was fit to the relevant patient data and the predicted parameter values were obtained. A
plethora of analyses was conducted on the model and parameter outcomes which included sensitiv-
ity analyses as well as an investigation into the dimensionless version of the model. The model was
found to be a suitable representation of the data although possible improvements were highlighted.
The model could capture the behaviour represented by the data well and the analyses proved that
a better understanding of the effects of the chemotherapy needs to be incorporated into the model.
Additionally, a patient specific method of parameter estimation would be more befitting due to the
specificity of each patient. A final aspect that could be investigated, is considering a version of the
model where the normal population within the model is separetely represented in the formulation.
This could enable further success with regard to model applicability.
Description
Keywords
Mathematical model, Gompertz model, Fractional calculus, Tumour growth, Rhabdomyosarcoma, Acute lymphoblastic leukaemia, Leucocytes, Lymphoblasts, Delay differential equations, Thesis (Ph.D. (Mathematics and Applied Mathematics))--University of the Free State, 2017