Using mathematics to weed out pesky tumours

The first stage of a typical tumour’s growth is called the avascular stage. At this point it possesses no blood vessels, and absorbs the nutrients needed for its growth from the inter-cellular fluid.

Gopikrishnan C R, a research scholar at the IITB-Monash Research Academy, is working on a project that does the modelling, numerical simulation and mathematical analysis of this stage of tumour growth. He is hopeful that his research will one day be able to save lives!

The project stands on three pillars: modelling of tumour growth in different circumstances, numerical simulations of the models, and mathematical analysis of the numerical methods employed to simulate the models.

Says Gopikrishnan, “What got me most interested in this project is that it links two mathematical communities, those who focus on the modelling part and those who do the analytical work. Both look at the same problem and understand the dynamics of tumour growth theoretically, using two different perspectives. The modelling community focuses on ‘how’, while the analysis community tackles ‘why’, and both are equally important.”

So how has his research progressed this far? “We have devised a method which addresses the moving boundary problem in tumour growth models. We have theoretically proved and illustrated the reliability and cost-effectiveness of the method. So, we now have a generic framework by which we can address tumour growth problems. This method has significant theoretical advantages as well. It helps answer deeper questions like whether the problem has a solution, and, if yes, whether our computer simulations correctly approximate the solution.”

But Gopikrishnan has no plans to stop here.

“Since we have developed a generic framework for basic tumour growth problems, we are now in a position to add complexities to the model. We can study the effect of an external cancer drug or about the depletion of nutrients or about developing blood vessels and how they pass on to the stage of malignancy. In a laboratory, testing all this takes many weeks and are costly in terms of money. But it can be reduced to hours if not minutes if we use using modelling and computer simulations.

“Basically, we observe the starting stage of a growing tumour and compare it with mathematically well-observed natural phenomena. A tumour is like a bunch of cells embedded in the intercellular fluid. In turn, the cells too behave like a fluid which is viscous than the intercellular fluid owing to its rough cell membrane. So a tumour can be imagined as mixture of two fluids, a viscous one and an inviscous one (see Figure 1).

Figure 1: Tumour cells and intercellular fluid     Figure 2: Exchange of material between cells and fluid

“A lot of research has been conducted in physics and mathematics on the theory of mixtures. Therefore, we model a tumour as a mixture of two fluids interacting with each other. The next question is: what are the important interactions? When the cells die its organelles disintegrate and become a part of the fluid and cells absorb fluid to divide and grow. In summary, cells and intercellular fluid constantly exchange matter with each other. This leads to a model based on mass conservation laws, which we numerically solve, study the solutions minutely, and then develop ways to improve the model.”

The IITB-Monash Research Academy is a collaboration between India and Australia that endeavours to strengthen scientific relationships between the two countries. Graduate research scholars like Gopikrishnan study for a dually-badged PhD from both IIT Bombay and Monash University, spending time at both institutions to enrich their research experience.

Prof Murali Sastry, CEO, IITB-Monash Research Academy, says, “We wish Gopikrishnan the very best. India loses approximately 700,000 lives every year to cancer. What could be better than saving some of these!”

Please click here to watch a one minute animation of Gopikrishnan’s thesis.

Research scholar: Gopikrishnan C R, IITB-Monash Research Academy

Project title: Numerical methods for free boundary problems in three dimensions with applications in biology

Supervisors:  A/Prof Jerome Droniou, Dr Jennifer Flegg, and Prof Neela Nataraj

Contact details:

This story was written by Mr Krishna Warrier based on inputs from the research student, his supervisors, and the IITB-Monash Research Academy. Copyright IITB-Monash Research Academy.