Cancer remains one of the major causes of death worldwide in spite of the large resources on research into therapies in the last 20 - 30 years. Preclinical tumour studies aim to identify potential drug combinations, which could be more efficacious. The results of these preclinical studies can then warrant further investigation in clinical trials, with the aim of confirming efficacy and translational relevance. Recent years have provided potential combination therapies such as radiotherapy (RT) in combination with DNA damage response inhibitors (DDRi) or immune checkpoint inhibition (ICI), but there is a very low amount of information regarding dosage and schedule optimisation strategies. Mathematical modelling can be used to identify optimal doses and schedules in the context of preclinical experiments as well as clinical studies. Current modelling of the impacts of radiation in combination with DDRi or ICI are limited to a few specific preclinical experiments, with little evidence of external validation. These models are developed from data collected during experiments in highly responsive xenograft or syngeneic tumour models that do not capture the response of the typical tumour in the clinic. Furthermore, at the beginning of the PhD project, while models of RT in combination with DDRi or ICI have been developed, there were no published models describing the impact of RT/DDRi/ICI in triple combination therapies. The overarching aim of this thesis was to develop mixed effects models that described the impact of RT/DDRi/ICI in the context of the syngeneic tumour model â MC38, in order to identify optimal doses and schedules of RT/DDRi/ICI to assist in preclinical experiments. In addition, these mixed effects models would also be used to validate potential biomarkers, which could confirm that the immune response drives tumour rejection after RT, or RT/DDRi in MC38 tumours, these biomarkers could then be used to assist in parameterisation for further model development. Preclinical studies involving MC38 syngeneic tumour models had indicated that at the given dosage and schedule, RT/DDRi/ICI had no observable benefit in tumour response compared with RT/ICI. The first aim of this project was to develop a mixed effects model, which captured the differential responses to RT/DDRi/ICI and corresponding de-escalated therapies. The mixed effects model was developed to capture the lack of observable benefit of RT/DDRi/ICI compared with RT/ICI, and then to simulate alternative potencies of RT/ICI which could show a relative improvement in efficacy of RT/ICI combined with AZD0156 (ATMi) compared with RT/ICI combinations. After successful validation of the model, this work revealed that reducing the potency of ICI by 68% could lead to an improved chance of observing the benefits of incorporating ATMi as part of a tritherapy. This would require additional experimental validation with MC38 models to identify the appropriate dosage. For further model development, it is important to identify appropriate, biologically relevant mechanisms that can be incorporated into the model. Mixed effects modelling strategies can assist in identifying drug mechanisms-of-action. These mechanisms can be parameterised in order to develop more complex mathematical models that more appropriately capture the pharmacodynamic effects of different treatment modalities on tumour control. However, the datasets available only contained data regarding tumour immunophenotypes at day 7. Thus, the next aim was to confirm whether the mixed effects model developed previously, could be used to correlate the expected growth rate of the tumour at day 7 (DD7) with survival time using a Cox regression model. A Cox regression analysis indicated that DD7 was strongly associated with survival time in MC38 syngeneic tumour models treated with RT. However, the relationship between DD7 and survival in mice treated with RT/DDRi was less predictive of response due to the lack of variability in DD7 estimates within this treatment arm. This indicated that a modified dosage or schedule of RT/DDRi would be required to increase the variability in response, in order to detect relationships between biomarkers and response to RT/DDRi. After a successful Cox regression analysis confirming that DD7 estimates correlated with survival in RT treated mice, the next aim was to identify candidate biomarkers implicated in the response to RT in MC38 mice by assessing which biomarkers correlated with DD7. At the time of writing, there are no validated clinical biomarkers that correlate with the response to radiation as a monotherapy. Part of the reason for this is the inter-patient variability in responses to radiation that may not be appropriately characterised with linear combinations of gene expression. To test the hypothesis that it could be possible to cluster different immunophenotypes and assess whether different clusters can be associated with the response, a hierarchical cluster analysis was performed to detect cluster dependant biomarkers from mice treated with RT. The results indicated a cluster dependant positive correlation between CD8-Ki67 fluorescence and DD7, as well as a cluster dependant relationship between the abundance of natural killer cells and DD7. These biomarkers were then validated as covariates within a mixed effects model. The results from these analyses indicated that T cell exhaustion was one of the major mechanisms of tumour relapse during RT, and that RT in combination with AZD7648 (DNAPKi) was successfully capable of minimising T cell exhaustion, and that this was the major reason for the lack of variability in tumour growth rates during RT/DNAPKi. Thus, the results of the cluster analysis and mixed effects modelling confirmed that cluster analysis may be an appropriate method to detect biomarkers within the context of radiation treated tumours. In addition, the data provided which confirmed that CD8+ T cell exhaustion played a role in immune escape after radiation was also deemed sufficient to parameterise a mathematical model that described the effect of radiation induced T cell exhaustion in MC38 cells. Thus, the next aim was to produce a model that successfully captured how RT/DNAPKi could minimise the rate of T cell exhaustion, using T cell exhaustion parameters calculated using the pharmacodynamic data available. The model successfully captured the lack of variability in tumour size to RT/DNAPKi, and was then used to simulate alternative schedules of RT/DNAPKi, which could be used to increase the variability in response, as well as find an appropriate time point of interest to assess biomarker data. Simulations of one day of RT/DNAPKi, followed by four days of RT as monotherapy, indicated that this schedule of therapy would sufficiently increase the variance in response in order to detect biomarkers implicated in the response to RT/DNAPKi. Given the negligible difference in efficacy between RT/ICI and RT/ICI in combination with olaparib (PARPi), the model was then applied to RT/PARPi/ICI data with the aim of quantifying the extent that ICI was contributing to the reduction in T cell exhaustion, and how this could be overshadowing the effects of PARPi in a tritherapy. The model was successfully validated on RT/PARPi/ICI data before simulating alternative dosages and schedules of RT/PARPi/ICI in order to determine appropriate dosages, which would implicate the role of PARPi in an RT/PARPi/ICI regimen. Simulations indicated that reducing the dose of ICI to between 2-4mg/kg could lead to a significant improvement in response rates between tritherapy and corresponding bitherapies. The previous model that was developed would have required additional experiments to confirm the appropriate dosage of RT/ICI to assess as part of a tritherapy. While the general conclusion of this work converged with the previous model that the dose of ICI needed to be reduced in order to parse the effects of DDRi and ICI in combination with RT, this model expanded on the previous model as it was able to provide an exact dosage and schedule to assess in preclinical experiments. These experiments will also assist in identifying biomarkers that cause the differential effects of RT/DDRi and RT/DDR/ICI easier to identify â to further elucidate the mechanisms, which can be parameterised in further models. One major issue with the models described within this thesis, as well as models describing radiation in combination with immunogenic drugs, is that the majority of these models predominantly focus on the immunogenic effects of RT, and how these immunogenic effects are perpetuated with additional drugs. Additional data is necessary in order to develop a model that can incorporate the effects of radiation induced cell death as well as immunogenic cell death. Magnetic resonance imaging (MRI) can provide additional information on the cellularity and oxygenation of a tumour and could provide potential to develop such a model. Thus, the thesis was refocused to observe whether mixed effects models could be used to develop a mathematical model that utilises MRI data. The MRI data available was able to describe changes in both cell density and tumour oxygenation, and thus the model aimed to capture how changes in density and oxygenation affected the response to various therapies, to then assess for optimal dosing regimens. The mixed effects model was validated with an internal dataset obtained from Calu6 xenograft models and the results indicated that administration of treatment holidays could be beneficial when attempting to maintain local tumour control. The results of this work could now be applied to immunocompetent tumour bearing mice in order to identify appropriate schedules linking both direct cell death, as well as immunogenic cell death on the response to immunogenic combination therapies, as well as to identify additional biomarkers implicated in the response to radiation, which can be parameterised within mixed effects models.