Despite clinical successes, data from trials of cancer immunotherapies continue to highlight non-responders. A better understanding of tumour-immune interactions is needed to explain outcome variability. In this thesis, we investigate whether tumour-immune interaction models can explain variation in mouse tumour volume time series data, focusing on a well-known mathematical model by Kuznetsov et al. (1994).
Exploiting a separation of timescales in the original model, and using matched asymptotics, we derive a new long-timescale approximation of the model, which differs from the quasi-steady-state approximation (QSSA) introduced by Kuznetsov et al. (1994), but is in good agreement with numerical solutions of the original model. As well as exhibiting the three stages of immunoediting – elimination, equilibrium and escape, together with complex bifurcations and bistability, our reduced model highlights excitability, a feature not traditionally associated with tumour-immune models. We also identify different biophysical parameters that could be targeted with immunotherapy in order to control tumour size.
As the QSSA model is simpler, but still exhibits immunoediting, we use it for case studies investigating parameter identifiability for complex bifurcating and bistable models using in silico data, generated from the model with variable parameters, observables and noise levels. Using profile likelihoods, we show that identifiability varies across dynamically different datasets. Parameters related to the immune system are highly correlated, and can result in large parameter confidence intervals and bimodal likelihood surfaces.
We then explore a population approach, where the model is simultaneously calibrated to dichotomous data from two individuals, noting that variation in tumour outcomes can arise from bistability or bifurcations, representing distinct hypotheses for heterogeneous tumour responses. We demonstrate the hypotheses can be distinguished for most synthetic datasets considered, except for highly noisy tumour data, by comparing Bayesian population fits under each hypothesis.
This thesis provides a new perspective on how qualitative differences between tumour growth dynamics of responders and non-responders may be explained by bifurcating and/or bistable models. At the same time, it illustrates how integration of mathematical and statistical methods can facilitate development of simplified models, their analysis and calibration.</p
