Practical approaches to the application of mathematical modelling in oncology: from model investigation to data-driven model construction
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.
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