# Community Research and Development Information Service - CORDIS

## New mathematical models for tumour growth

Mathematicians are developing new algorithms for modelling complex physical and biological two-phase systems such as turbulent gas flows, melting-evaporation processes and tumour growth.
Processes in which matter transitions from one phase to another (e.g. solid to liquid when ice melts) can be mathematically modelled using partial differential equations (PDEs). The heat equation, for example, is a PDE describing the distribution of heat (or variation in temperature) in a given region over time.

When transitioning states have moving boundaries or interfaces that separate the two phases, the resulting discontinuities across interfaces complicate mathematical modelling. For example, a block of ice just starting to melt has a liquid-solid phase boundary with a varying, unknown temperature distribution over time. To solve this so-called boundary value problem (BVP), mathematicians apply the Stefan boundary condition that accounts for movement of unknown boundaries.

The EU-funded BVPSYMMETRY (Reductions and exact solutions of boundary value problems with moving boundaries by means of symmetry based methods) project aimed to find solutions to BVPs using new algorithms, and to apply these algorithms in physical and biological models.

Since complex biological and physical systems often behave in inconsistent ways, they are difficult to model using normal mathematical equations. Using Lie symmetry analysis, BVPSYMMETRY reduced BVP equations into their normal form to find exact solutions to complex 2D and 3D problems.

To demonstrate their algorithms, researchers tested them in heat transfer and population dynamics models and then applied their theoretical foundations to more complicated BVPs. These included multidimensional problems and those with moving boundaries, such as models for turbulent gas flows.

Models of tumour growth can be described using mathematical models incorporating BVP with moving boundaries In this case the equations incorporate diffusion, proliferation and spread of tumour cells, the rates of which are not constant and depend on multiple factors. BVPSYMMETRY simplified the multidimensional BVP of tumour growth to develop a 1D model that may also be applied to other complex multidimensional systems.

Future models may also incorporate killing rates of cells into the equations, allowing doctors to predict how effective different treatments are likely to be.

## Subjects

Scientific Research

## Keywords

Mathematical models, tumour growth, PDE, moving boundaries, BVP, Lie symmetry analysis
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