Cancer development models with extinction threshold
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Date
2025-08-18
Authors
Bastian, Frank
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Publisher
University College Cork
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Abstract
The central idea of this thesis is to include the immune system in a conceptual cancer development model as an extinction threshold, similar to the strong Allee effect in population biology.
We start by reviewing classical population growth models.
Based on this review, we identify the limitations of commonly used Allee effect models in reproducing typical cancer progression.
We then address these limitations by deriving a new model that incorporates:
(i) random mutations of stem cells at a rate that increases with age and
(ii) immune response whose strength may also vary over time and propose a simple dynamic model of cancer development that captures carcinogenesis and subsequent cancer progression.
Our model accurately reproduces a wide range of real-world cancer data:
the typical age-specific cumulative risk of most human cancers, the progression of transplanted breast cancer tumours in mice, and the unusual age-specific cumulative risk of breast cancer in women. In the last case, we use a moving extinction threshold to reflect the different strengths of the immune response at different phases of the menstrual cycle and menopausal treatment. This provides new insights into the effects of hormone replacement therapy and menstrual cycle length.
This moving threshold approach can be applied to a variety of other cancer scenarios where the immune response or other important factors may vary over time.
Cancer development and treatment is often modelled as a coupled multi-population system consisting of treatment-resistant and sensitive cancer cell populations that can either cooperate or compete for resources. This problem is often viewed through the lens of {\em evolutionary game theory}, where generalised Lotka-Volterra equations for the dynamics of the populations are reduced to so-called replicator equations with linear payoffs for the dynamics of the proportions of the total population that use different strategies.
Here, we revisit the equivalence relation between multi-population and replicator dynamics. In addition, we propose an extension to non-linear (state-dependent) payoffs, which arise naturally in the presence of an extinction threshold, leading to a generalised replicator equation.
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Keywords
Cancer development , Mathematical model , Extinction threshold , Breast cancer risk , Allee effect , Noise-induced tipping
Citation
Bastian, F. T. 2025. Cancer development models with extinction threshold. PhD Thesis, University College Cork.