Multiphase Tumour Growth 1-Dimensional Slab Model

Research question

How can a two-phase fluid model describe tumour growth in a one-dimensional slab, and what does the simplified model reveal through stability analysis and numerical solutions?

Background

The project models a tumour as a mixture of cells and water with nutrient transport and mechanical effects. It was completed as an extra Year 2 summer mathematics department project with Paarth Saini, Anson Hui, and Aziz Zorlu.

Mathematical model

The full model uses cell and water volume fractions, velocities, pressures, stress tensors, nutrient concentration, cell production, and nutrient consumption. The no-voids assumption reduces the volume fractions to a constrained mixture. The project then simplifies the system to a one-dimensional slab geometry so that stability and numerical behaviour can be studied more directly.

Methods

The report is organised around model derivation, reduction to a one-dimensional geometry, stability analysis, numerical solution, and critique. This structure makes the modelling assumptions visible before interpreting numerical results.

Implementation

The public PDF is stored as a report artifact. Future code links should only be added after the notebooks or scripts are cleaned, documented, and checked against the equations in the report.

Results

The report gives a coherent derivation of the variables and equations, then uses analysis and numerical solutions to explore how the tumour-growth model behaves under the selected assumptions.

Validation

The main validation route is internal consistency: checking assumptions, boundary conditions, limiting cases, and whether numerical behaviour agrees with the reduced model. The project also includes a critique section so the model is not presented as a complete biological description.

Limitations

The one-dimensional slab geometry is a strong simplification. The model should be read as a mathematical and computational study rather than a clinical or biological prediction tool.

What I learned

The project connects PDE modelling, continuum assumptions, stability analysis, numerical computation, and scientific communication in a group-report setting.

Future work

Future work could include cleaned code, parameter sensitivity, comparison with higher-dimensional geometries, and clearer links between the mathematical assumptions and biological interpretation.