Oxygen-Driven Tumour Growth Model: A Pathology-Relevant Mathematical Approach

Xenografts -as simplified animal models of cancer- differ substantially in vasculature and stromal architecture when compared to clinical tumours. This makes mathematical model-based predictions of clinical outcome challenging. Our objective is to further understand differences in tumour progression and physiology between animal models and the clinic.To achieve that, we propose a mathematical model based upon tumour pathophysiology, where oxygen -as a surrogate for endocrine delivery- is our main focus. The Oxygen-Driven Model (ODM), using oxygen diffusion equations, describes tumour growth, hypoxia and necrosis. The ODM describes two key physiological parameters. Apparent oxygen uptake rate (kR′) represents the amount of oxygen cells seem to need to proliferate. The more oxygen they appear to need, the more the oxygen transport. kR′ gathers variability from the vasculature, stroma and tumour morphology. Proliferating rate (kp) deals with cell line specific factors to promote growth. The KH,KN describe the switch of hypoxia and necrosis. Retrospectively, using archived data, we looked at longitudinal tumour volume datasets for 38 xenografted cell lines and 5 patient-derived xenograft-like models.Exploration of the parameter space allows us to distinguish 2 groups of parameters. Group 1 of cell lines shows a spread in values of kR′ and lower kp, indicating that tumours are poorly perfused and slow growing. Group 2 share the value of the oxygen uptake rate (kR′) and vary greatly in kp, which we interpret as having similar oxygen transport, but more tumour intrinsic variability in growth.However, the ODM has some limitations when tested in explant-like animal models, whose complex tumour-stromal morphology may not be captured in the current version of the model. Incorporation of stroma in the ODM will help explain these discrepancies. We have provided an example. The ODM is a very simple -and versatile- model suitable for the design of preclinical experiments, which can be modified and enhanced whilst maintaining confidence in its predictions.Author Summary: Tumour-bearing animal models of cancer are needed to discover new drugs to treat cancer. We aim in this article to capture—through mathematics- some underlying phenomena of tumour growth in animals. We propose a set of equations that, despite being very simple, describe tumour growth, hypoxia and necrosis. Cells under low oxygen levels change into a stress state called “hypoxia”, which will ultimately lead to tissue death, also known as “necrosis” and “apoptosis”. Hypoxic cells undergo a variety of changes which impact tumour growth, development, metastasis and -most importantly- response to therapy. Hence, oxygen distribution is important. We simulate oxygen profiles to locate hypoxic and necrotic tumour regions. Finally, this mathematical model allows us to compare and classify animal models from a grounded and physiological perspective, rather than a more convenient and empirical one. This will help us understand how well (or poorly) animal tumours mimic tumours in patients. The simplicity of our mathematical model allows us to obtain more information out of the same animal sets without any further experiments, hopefully saving time, money and animal usage.